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.

Filter by
Sorted by
Tagged with
7
votes
0answers
82 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, ...
2
votes
1answer
106 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 $\...
-4
votes
1answer
308 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 $...
0
votes
0answers
262 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 ...
24
votes
1answer
718 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 ...
6
votes
2answers
157 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.
7
votes
2answers
229 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 ...
3
votes
2answers
184 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. ...
8
votes
2answers
217 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 ...
2
votes
0answers
87 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 ...
0
votes
1answer
57 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}} \...
2
votes
0answers
58 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^...
1
vote
1answer
119 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 ...
2
votes
1answer
206 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+\...
0
votes
1answer
146 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 ...
8
votes
1answer
369 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 ...
14
votes
1answer
371 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&...
33
votes
1answer
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-...
3
votes
0answers
94 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 ...
0
votes
1answer
748 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 ...
4
votes
0answers
89 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)\...
6
votes
1answer
268 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)}-\...
20
votes
1answer
657 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 ...
3
votes
0answers
121 views

L functions of Symmetric power of elliptic curves

Let $E$ be an elliptic curve over the raional field with conductor $N$, which corresponds to the eigenform $f(z)=\sum a_nq^n$. Let $L(Sym^2E,s)$ be the L function of the symmetric power of $E$.I am ...
1
vote
0answers
73 views

Rankin-Selberg method and Symmetric power of elliptic curves

Let $E$ be an elliptic curve over $\mathbb{Q}$ with conductor $N$. Let $f=\sum a_n q^n$ be the weight 2 modular form corresponding to $E$. Define $L_2(f,s)=\zeta(s-1)L(Sym^2(E),s)$. The following ...
8
votes
1answer
293 views

Proof of a 'well-known' result of Shimura on periods of modular forms

It is often noted in the literature that there are certain complex periods that allow one to normalize the modular symbol associated to a modular form in such a way that its coefficients are algebraic....
3
votes
0answers
85 views

L functions of elliptic curves over quadratic fields

Let $E$ be an ellitpic curve over a quadratic field $K/\mathbb{Q}$. Then the L function of $E$ is defined as $L(E_K,s)=\prod_{\mathfrak{p}\nmid \Delta}(1-a_{\mathfrak{p}}N(\mathfrak{p})^{-s}+N(\...
3
votes
0answers
107 views

On the ''generalised'' Chebyshev psi function

Let $\chi$ be a Dirichlet character mod $q$ and $\Lambda(n)$ be the von Mangoldt function. Let $c(\chi)=1$ if $\chi$ is the principal character, and zero otherwise. Let $\Theta_\chi$ be the supremum ...
1
vote
0answers
225 views

On the error bound for the Prime Number Theorem for arithmetic progressions

Let $\chi$ be a Dirichlet character, $L(s,\chi)$ be the corresponding L-functions and $\Theta_{\chi}$ be the supremum of the real parts of the zeros of $L(s, \chi)$. Define $\pi(x; a, q)$ to be the ...
4
votes
0answers
79 views

Honda-Tate theorem and prescribing roots of $L$-functions

I'm currently working with Artin $L$-functions in function field extensions (on one variable, over finite fields). I have heard about a theorem of Honda-Tate which vaguely states that a polynomial $P \...
5
votes
0answers
556 views

Twisted functional equations in Goldfeld's book

I am confused about the contents of D. Goldfeld's book "Automorphic forms and L-functions for the group $\mathrm{GL}(n,\mathbb{R})$". In the process of deriving the converse theorem on $\...
1
vote
1answer
123 views

Moments of Dirichlet $L$-functions on the critical line

I'm looking for a reference for some questions related to the moments of Dirichlet characters on the critical line, $$ M_k(T;\chi) = \int_T^{2T} |L(1/2+it,\chi)|^{2k}\,dt, $$ where $\chi$ is a ...
5
votes
0answers
144 views

Non-vanishing of L-values for twists of Hilbert modular forms

Let $F$ be a real quadratic field, and let $f$ be a Hilbert modular form over $F$ of parallel weight 2. It's known, by a theorem of Rohrlich, that there exist infinitely many Hecke characters of $F$ ...
0
votes
1answer
199 views

On $L$-function of permutation representation

I came across the statement in a book: Let $k$ be a number field and $K$ be a Galois extension of $\mathbb Q$ containing $k$, with Galois group $G=\operatorname{Gal}(K/\mathbb Q)$ and let $G_k:=\...
0
votes
1answer
94 views

Logarithms of $L$-functions of irreducible characters of Galois group

We know that the $L$ functions of Dirichlet characters $\chi$ of $(\mathbb Z / m\mathbb Z)^\times$ satisfy the property that $\log L(s, \chi)$ is holomorphic for $\Re(s) \geq 1$ if $\chi$ is a ...
3
votes
1answer
139 views

English reference for the Brauer-Kuroda formula

I'm currently trying to understand the Brauer-Kuroda formula. Although there are many recent papers on the formula but they seem to be purely algebraic. They say that original analytic approach is ...
2
votes
0answers
103 views

The definition of Langlands' L-function $L(s,\pi,r)$ in the case of $\operatorname{GL}_1$

Let $G$ be a split reductive group over a $p$-adic local field $k$. For $\pi$ an unramified representation of $G(k)$, and $r$ a finite dimensional representation of the L-group $^LG$, Langlands ...
6
votes
2answers
435 views

Analytic equivalents for primes in arithmetic progressions

By way of context: it is known that the prime number theorem $\pi(x) \sim x/\log x$ is (nontrivially) equivalent to the statement that $\zeta(s)$ does not vanish on the line $\Re s=1$. I would like ...
4
votes
0answers
109 views

Small values of $L(\frac{1}{2}, \chi_{-8d})$

In Section 3 of https://arxiv.org/pdf/0708.3990.pdf Soundararajan shows that there exists infinitely many fundamental discriminants $8d$ for which $L(1/2, \chi_{8d})$ is very small. His argument runs ...
1
vote
1answer
197 views

Under Ramanujan conjecture, is primitivity equivalent to cuspidality and irreducibility?

Lemma 4.2 in M. Ram Murty, Selberg conjectures and Artin L-functions(1994), states that under Ramanujan conjecture, an irreducible cuspidal automorphic representation of $\operatorname{GL}_{n}(\mathbb{...
3
votes
3answers
506 views

Functional derivatives on Banach spaces

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...
3
votes
0answers
341 views

Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?

To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$. Is the map $\pi\mapsto L_{\pi}$ bijective? Edit March ...
2
votes
0answers
68 views

Relations between spectral parameters of automorphic representations

Let $\pi$ be an automorphic representation (say, trivial central character) of $GL(2)$. Let $\alpha(p)$ and $\beta(p)$ denote its spectral parameters at the place $p$, that is to say the associated ...
6
votes
1answer
219 views

“Sub-logarithmic” zero-free regions from Deuring-Heilbronn/Linnik's repulsion theorem

For each $n\in\mathbb{N}$, let: $\chi_n\pmod{q_n}$ a real non-principal Dirichlet character ($q_1 < q_2 < \cdots$), $\beta_n$ the largest real zero of $L(s,\chi_n)$, $\delta_n := (1-\beta_n)\...
4
votes
1answer
209 views

Is the imaginary part of $t\mapsto\zeta(1/2+it)$ close to the derivative of its real part?

Plotting $t\mapsto\zeta(1/2+it)$ on Wolfram alpha, it seems that the maxima of its real part are close to the zeros of its imaginary part, while the maxima of the latter seem close to the inflection ...
3
votes
0answers
113 views

Proving that central zero of an L-function is multiple

Central point of critical strip, ie. $s=1/2$, is conjectured to be the only argument at which an L-function can have multiple zero. This is interesting ia. for proving effective lower bounds for class ...
9
votes
1answer
357 views

Are the L-functions of a normalized newform and the corresponding cuspidal representation equal?

Let $f \in S_k(\Gamma_0(N))$ be a normalized newform with Fourier expansion $$f(z) = \sum\limits_{n=1}^{\infty} a_n e^{2\pi i z n}$$ and $a_1 = 1$. Then $f$ is an eigenfunction of all Hecke ...
8
votes
1answer
406 views

Does the symmetric square L-function vanish at one?

Take a cuspidal automorphic representation $\pi$ of $GL(3)$ over a number field. My question is quite straightforward and can be related to this one : Can $L(1, \pi, \mathrm{sym}^2)$ be zero? If ...
5
votes
1answer
294 views

Proving automorphy of the Galois representations of number fields without considering the residual representation

All the papers proving automorphy of the representations of Galois groups of number fields that I have come across seem to first reduce the representation modulo a prime, prove the automorphy of the ...
2
votes
2answers
235 views

Upper bound of summation $\sum_{m < \frac{1}{2}X} \frac{|a(m_1m_2^2)|}{m_1m_2^2} \log\frac{X}{m}$

I am studying the paper M. Ram Murty, V. Kumar Murty: Mean values of derivatives of modular $L$-series, Ann. of Math. (2) 133 (1991), no. 3, 447-475. Let $L(s)=\sum_{n=1}^{\infty} \frac{a(m)}{m^s}$ ...

1
2 3 4 5
7