Questions tagged [l-functions]

Questions about generalizations of the Riemann Zeta function of arithmetic interest whose definition relies on meromorphic continuation of special kinds of Dirichlet series, such as Dirichlet L-functions, Artin L-functions, elements of the Selberg class, automorphic L-functions, Shimizu L-functions, p-adic L-functions, etc.

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The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$

Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...
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-4 votes
0 answers
83 views

Does any element of an L-rig fulfill Rudnick and Sarnak hypothesis H and thus Selberg orthogonality conjecture?

I introduce the notion of L-rig in the first paragraph of Are there infinitely many L-rigs? Calling "genuine L-function" any element of an L-rig, and to the light of recent results about L-...
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9 votes
1 answer
265 views

Is $\frac{1}{L(1+it)}$ unbounded?

Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\frac{1}{L(1+it, \chi)}$$ unbounded for $t \in \mathbb{R}$? I'm aware that this is true if $L=\zeta$, but I'm ...
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2 votes
1 answer
94 views

Do Artin L functions have polynomial growth in in the critical strip?

Given an irreducible representation $\rho$ of the Galois group $G$ of a number field $K$ over $\mathbb{Q}$, we have the associated Artin $L$ function which we denote by $L(s, \rho)$. By Brauer ...
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1 vote
0 answers
41 views

Explanation about Lapid-Rallis iductive argument (doubling method)

I am reading Lapid-Rallis "On the local factors of representations of classical groups" and I am completely stuck with the proof of Proposition 3. In the case $\mathcal V$ is not anisotropic,...
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2 votes
0 answers
122 views

Symmetric square L-function with non square-free level

Let $f$ be a primitive holomorphic cusp form of weight $k$, level $N$ and nebentypus $\chi$, with its $L$-function $L(s,f)=\displaystyle\sum_{n\geq1}\lambda_f(n)n^{-s}$ for $\mathrm{Re}(s)>1$. Let $...
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1 vote
0 answers
128 views

Large values of $L(1,\chi)$ for quadratic Dirichlet characters $\chi$

Granville and Soundararajan, in "Upper Bounds for $L(1, \chi)$", first paragraph, say it is known that there exist quadratic Dirichlet characters $\chi$ for which $L(1, \chi)$ is about $\log\...
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3 votes
0 answers
201 views

Have there been recent developments of Booker's approach to L-functions as distributions?

Andrew Booker introduced a framework to study L-functions through distributions in https://arxiv.org/abs/1308.3067v2. This allowed him and others to get new results about zeros of automorphic L-...
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2 votes
2 answers
169 views

Sign of the special value at s=0 of Hecke L-functions

Let $L/K$ be an abelian extension of number fields with Galois group $G$ and let $\chi : G \to \{\pm 1\}$ denote a real linear character of $G$. Denote $L(\chi,s)$ the Artin L-function associated to $\...
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3 votes
0 answers
71 views

Hoffstein–Lockhart for non-congruence subgroups

Let $\Gamma$ be a non-congruence subgroup of $\operatorname{SL}(2,\mathbb{Z})$ of finite index and let $f$ be a holomorphic cuspidal modular form of weight $k$ for the group $\Gamma$. For simplicity, ...
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2 votes
1 answer
113 views

$p$-adic valuation of $L$ values for elliptic curves

I'm wondering if the following conjecture is true: Let $\mathcal{A}$ be an isogeny class of elliptic curves over $\mathbf{Q}$. Fix an odd prime $p$ of good reduction. Then there is a curve $E \in \...
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  • 695
1 vote
0 answers
75 views

Relation between $L$-values of elliptic curves and Manin constants

Given an elliptic curve $E$ over $\mathbf{Q}$, we can attach two numbers two it. the so-called Manin constant $c_E$. (Defined below the fold.) the "algebraic $L$-value" given by $L(E,1)/\...
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  • 695
9 votes
2 answers
353 views

Computing the Petersson norm of newforms of weight 2 from the symmetric square $L$-function

Let $f \in S_2(\Gamma_0(N))$ be a newform with trivial character. I want to compute the Petersson norm $\lVert f\rVert^2$ of $f$, not normalized by $1/[\operatorname{SL}_2(\mathbf{Z}):\Gamma_0(N)]$, ...
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7 votes
1 answer
277 views

Explicit estimates for $N(T,\chi)$ (not $N(T,\chi)+N(T,\overline{\chi})$)

Let $N(T,\chi)$ denote the number of zeros of $L(s,\chi)$ with imaginary part between $0$ and $T$, with any zero with imaginary part equal to $T$ or to $0$ (not that the latter kind really exists) ...
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1 vote
0 answers
105 views

Is it possible $L(\frac{1}{2},\phi \times \phi')=0$ for all $\phi'$?

Let $\phi$ be an irreducible cuspidal automorphic representation of $GL_n(\mathbb{A})$ of symplectic type, that is, the exterior square $L$-function $L(s,\phi,\Lambda^2)$ has a pole at $s=1$. Then I ...
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  • 1,525
0 votes
1 answer
160 views

Are Li's numbers $\lambda_n$ absolutely convergent for $n>1$?

Li's numbers $\{\lambda_n\}$ are defined as $$\lambda_n=\frac{1}{(n-1)!}\frac{d^n}{ds^n} [s^{n-1}\log\xi(s)]_{s=1} $$ for all positive integers $n$. Also $\lambda_n$ is given as a sum over the non ...
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4 votes
0 answers
81 views

Sign error in $\pm$-parts of modular symbols?

I am trying to connect the definition of $\pm$-modular symbols given in [Pollack, pg. 529] and [MTT,pg. 11] to those appearing in [Greenberg-Stevens, pg. 200 in #20 here], but I can't seem to ...
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  • 325
4 votes
0 answers
271 views

Automorphisms of the ring of periods

The set of periods $\mathcal{P}$ introduced by Kontsevich and Zagier forms a ring, see for example https://en.m.wikipedia.org/wiki/Period_(algebraic_geometry). Moreover J. Wan introduced in 2011 in ...
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4 votes
1 answer
947 views

Does the existence of a Landau-Siegel zero imply the existence of a complex zero off the critical line?

The question is in the title: can a Landau-Siegel zero be the only zero off the critical line for a Dirichlet L-function or does its existence imply the existence of a complex non trivial zero in the ...
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6 votes
0 answers
205 views

Why are the $p$-adic $L$-functions for a modular form with $a_p=0$ conjugates?

I have a question about the proof of Theorem 3.5 in Pollack's 2003 paper On the $p$-adic L-function of a Modular Form at a Supersingular Prime. The setup is as follows. Fix an eigenform $f\in S_k(N,\...
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  • 325
-3 votes
1 answer
185 views

Structure of the automorphism group of an L-rig

This question is a follow-up to Are there infinitely many L-rigs? which is already pretty convoluted. Define the $\varphi$-evaluation morphism at a complex number $s$ as $\epsilon_{\varphi,s}:F\mapsto ...
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2 votes
0 answers
133 views

Reference request for an English translation of a book of Tate

In this ongoing program, Professor Mahesh Kakde said that the best reference for learning about Stark and Gross-Stark conjecture is this book of John Tate. But this book is in French. Is there any ...
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1 vote
0 answers
72 views

The pole of symmetric square $L$-function of $GL(n)$ at $s=1$

Let $\pi$ be an irreducible cuspidal automorphic representation of $GL(n)$. Suppose the symmetric square $L$-function of $\pi$ $L(s,\pi,Sym^2)$ has a pole at $s=1$. Then since $L(s,\pi \times \pi)=L(s,...
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  • 643
3 votes
1 answer
97 views

How to compute period polynomial of a meromorphic cuspform explicitly?

I am looking for an algorithm to compute the period polynomial $$P(z,f) := \int_C f(\tau) (z-\tau)^{k-2} d \tau$$ for a cusp form $f(\tau)$ of weight-k, where $C$ is a path connecting $\tau =0$ and $\...
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3 votes
0 answers
76 views

Study of relative class number of 'non-abelian' CM field by using L-functions

I'm currently interested in finding good upper bounds for the relative class numbers of non-abelian CM-fields. So I'm looking for some references to learn the techniques that can be useful. So far, I ...
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  • 951
1 vote
1 answer
253 views

Behaviour of a certain $L$ function at $s=1$

I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a ...
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  • 444
8 votes
0 answers
169 views

Unexpected patterns on the graph of an L-function on the critical line

Let $L(s)$ be the $L$-function associated to the (only) classical modular form of weight $26$ and level $1$. The completed L-function $\Lambda(s)=2(2\pi)^{-s}\Gamma(s) L(s)$ is symmetric with respect ...
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  • 7,992
7 votes
0 answers
116 views

Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?

The Selberg class of $L$-functions are Dirichlet series $$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$ satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
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3 votes
1 answer
146 views

Watson's triple product for automorphic forms shifted by Maass rising operators

Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...
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  • 143
-4 votes
1 answer
352 views

Scaled Riemann zeta function with no zero in the critical strip

Update: I added $exp[i\theta_k(s)]$ in the definition of $\eta^*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues. Prime numbers are denoted as $...
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0 votes
0 answers
276 views

Is $\operatorname{Aut}(\mathcal{M})$ a fundamental group in Grothendieck's sense?

This question is a follow-up to Are there infinitely many L-rigs? and to Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?. I copy paste a deepl ...
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26 votes
1 answer
807 views

Relation between Schanuel's theorem and class number equation

(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation) It was recently brought to my attention that there ...
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6 votes
2 answers
225 views

Vinogradov-Korobov for Dirichlet L-functions?

Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
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  • 16.7k
6 votes
2 answers
267 views

Functional equation and/or growth estimates for a shifted L function

Consider the $L$-series defined by $$L_{\alpha,\chi}(s) = \sum_{n\geq 1} \frac{e^{2\pi i \alpha \Omega(n)} \chi(n)}{n^s} = \prod_p \left(1 - \frac{e^{2\pi i \alpha} \chi(p)}{p^s}\right)^{-1}.$$ It ...
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  • 16.7k
3 votes
2 answers
219 views

Distribution of zeros of real quadratic Dirichlet L-functions in small intervals

Motivation: Some data gathered on least quadratic nonresidues indicate that the zeros of quadratic Dirichlet L-functions are more evenly spaced than that in general Dirichlet L-functions. Question. ...
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  • 7,992
8 votes
2 answers
266 views

Do odd-weight cusp forms have analytic rank 0?

Let $f(z)=\sum_{n\ge 1}a_nq^n$ be a cusp form, where $q=e^{2\pi i z}$. Let $ L(s) = \sum_{n\ge 1} a_nn^{-s} $ be its corresponding L-function. The completed L-function of $L(s)$, $\Lambda(s)$, should ...
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  • 7,992
2 votes
0 answers
99 views

Iwasawa theory over function fields - How do eigenvalues vary in $\mathbb Z_\ell$ towers?

Consider a tower of curves $\dots \to C_n \to C_{n-1} \dots \to C_1$ over $\mathbb F_q$ where $C_n: f(x^{\ell^n},y) = 0$.We can look at the multiset $S_n$ of eigenvalues of the Frobenius $\sigma_q$ on ...
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  • 6,834
0 votes
1 answer
62 views

Extending functional inequality from rectangles to parallelograms

Let $f$ be a function defined on the unit square $R = [0,1]^2 \subseteq \mathbf{R}^2$ satisfying $f \geq 0$, $f(0,0) = 0$, $\frac{\partial{f}}{\partial{x}} \geq 0$, $\frac{\partial{f}}{\partial{y}} \...
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2 votes
0 answers
71 views

The Guinand-Weil explicit formula for Hecke characters

The Guinand-Weil formula for the Riemann zeta function is \begin{aligned}&\Phi (1)+\Phi (0)-\sum _{\rho }\Phi (\rho )\\={}&\sum _{p,m}{\frac {\log(p)}{p^{m/2}}}{\Big (}F(\log(p^{m}))+F(-\log(p^...
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  • 7,992
1 vote
1 answer
140 views

How to find an explicit value of a Hecke L-function using Magma?

I'm trying to compute special values of Hecke L-function for the field $K=\mathbb{Q}(\sqrt[5]{1})$ using Magma (more exactly, I need $L(k, \chi^k)$, $k$ - integer, $\chi$ - Hecke character for the ...
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  • 13
1 vote
1 answer
218 views

What's the motivation for the $3$ appearing in Iwaniec and Kowalski's definition of the analytic conductor?

In their book Analytic Number Theory, Iwaniec and Kowalski, on page 95, define the analytic conductor by the following formula: $\displaystyle{{\frak{q}}_{\infty}(s)=\prod_{j=1}^{d}\left(\vert s+\...
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0 votes
1 answer
155 views

Why does LMFDB refer to L functions having coefficients of type $a_p-a_{p^2}$ instead of just $a_{p^2}$?

Today I was reading LMFDB (the L-functions and Modular Forms DataBase), and I came across something that confused me. When discussing degree 3 L functions on this page, they assert that all the ones ...
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  • 2,187
9 votes
1 answer
482 views

Spectral decomposition of product of modular functions

The eigenfunctions of the Laplacian on $SL(2,\mathbb Z)\backslash \mathbb H$ are known to be given by three types: the constant function, the real analytic Eisenstein series (which come in a ...
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14 votes
1 answer
424 views

Bound for $GL(3)$ symmetric square

Let $\pi$ be an automorphic representation of $GL(3)$ over a number field. Let $a_n$ be the coefficients of $L(s, \pi, \mathrm{sym}^2)$. Do we know if $$\sum_{n>0} \frac{|a_n|}{n^s}$$ and $$\sum_{n&...
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35 votes
1 answer
2k views

The modularity theorem as a special case of the Bloch-Kato conjecture

In the homepage for the CRM's special semester this year, I found the interesting statement that the modularity theorem (formerly the Taniyama-Shimura-Weil conjecture) is a special case of the Bloch-...
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  • 2,371
3 votes
0 answers
133 views

Maass--Selberg for any Eisenstein series on higher rank

Does there exist a Maass--Selberg relation for any Langlands Eisensein series on $\mathrm{GL}(n)$? By any I mean an Eisenstein series which is induced from any standard parabolic with any discrete ...
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2 votes
1 answer
932 views

Are there infinitely many L-rigs?

$\DeclareMathOperator{\Q}{\mathbb{Q}}$Call "L-rig" any class $\mathcal{L}$ of L-functions of automorphic representations of $\operatorname{GL}_{n}(\mathbb{A}_{\Q})$ for some $n$ belonging to ...
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4 votes
0 answers
126 views

Lower bound on symmetric square L-function

In a paper of Soundararajan, equation (16c) states that for any Hecke eigenform $f$ of weight $k$, the symmetric square L-function at $1$ satisfies the bound $$(\log k)^{-2}\ll L(1, \mathrm{sym}^2f)\...
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6 votes
1 answer
359 views

Which cases of Beilinson-Bloch-Kato for elliptic motives are known?

Let $V$ be a semisimple geometric Galois representation of a number field. Then the Bloch-Kato conjectures state that $$ \operatorname{ord}_{s=0}{L(V^*(1),s)} = \operatorname{dim}{H^1_f(G_k,V)}-\...
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  • 13.9k
20 votes
1 answer
749 views

Hadamard factorization of L-functions

I have already asked this question here in a different form, but really need an answer. Let $L(s)$ be a "standard" $L$-function, say with Euler product, functional equation, etc... (Selberg ...
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