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.
435
questions
10
votes
1
answer
618
views
Why $p$-adic measures?
I'm currently learning about the Kubota–Leopoldt $p$-adic $L$-function and I'm noticing that many people view the Kubota–Leopoldt $p$-adic $L$-function as a measure as opposed to a $p$-adic analytic ...
- 1,869
2
votes
2
answers
235
views
Special values of non-cm $L$-functions
For the sake of simplicity, assume $f$ is a non-cm eigenform of weight $k$ on the group $\mathrm{SL}(2, \mathbb{Z})$. Are there any known results or conjectures regarding any special values of the ...
- 505
26
votes
4
answers
3k
views
Why do we care about the eigenvalues of the Frobenius map?
The Riemann hypothesis for finite fields can be stated as follows: take a smooth projective variety X of finite type over the finite field $\mathbb{F}_q$ for some $q=p^n$. Then the eigenvalues $\...
- 1,768
4
votes
0
answers
444
views
Question about a paper by Franca and LeClair in analytic number theory
I am reading an article "Transcendental equations satisfied by the individual
zeros of Riemann $\zeta$, Dirichlet and modular
L-functions" by G. Franca and A. LeClair (2015) see here. The ...
- 41
3
votes
0
answers
167
views
Cohomology and p-adic L-functions
The definitions of $p$-adic $L$-functions I know all are given by interpolating the values of "usual" $L$-functions at the negative integers. If I want to define the $p$-adic $L$-function of ...
- 4,246
2
votes
2
answers
299
views
Reference for zero sum estimates of Dirichlet L functions
Let $\chi$ be a primitive character mod $p$ (prime) and $\rho = \beta + i \gamma$ be a non-trivial zero of $L(s, \chi)$.
I am reading a paper by Ihara and Murty where they use following estimate :
$\...
- 41
0
votes
1
answer
191
views
Spacings of Satake parameters under Ramanujan conjecture
I would like to know if, under Ramanujan conjecture, the following three distributions are known or conjectured to match:
the distribution of spacings between Satake parameters of an L-function $F$ ...
- 6,932
5
votes
2
answers
304
views
Additivity of Elliptic Curve Rank over Compositum of Fields
Assume that BSD holds for number fields. Let $E/\mathbf{Q}$ be an elliptic curve. For simplicity, let's assume it has Mordell-Weil rank zero. Let $F_1/\mathbf{Q}$ and $F_2/\mathbf{Q}$ be finite, ...
- 1,412
2
votes
1
answer
714
views
Does asymptotic Goldbach imply GRH?
It seems to me that a proof of $\alpha_{n}=o(n)$ where the quantity $\alpha_{n}$ is defined in About Goldbach's conjecture together with the main result of https://kyushu-u.pure.elsevier.com/en/...
- 6,932
-2
votes
1
answer
219
views
Special value of Hecke $L$ function
Let $E:y^2＝x^3-x/ \Bbb{Q}(i)$ be elliptic curve and $L(E,1)$ be a special value of $L$ function of $E$ at $1$.
Let $L(ψ,1)$ be value at $1$ of Hecke $L$ function with respect to Hecke character $ψ$, ...
- 1,405
3
votes
0
answers
216
views
Proof of $L(E,1)/Ω(E)=1/8$ for elliptic curve $E:y^2=x^3-x/ \Bbb{Q}$?
Let
$E:y^2=x^3-x$ be an elliptic curve over $ \Bbb{Q}$ and
$ω_E=dx/2y=dx/2\sqrt{x^3-x}$.
Then
$$
\begin{split}
\Omega(E)&=\int_{E(\Bbb{R})} ω_E\\
\\
&=2\int\limits_1^{+\infty} dx/\sqrt{x^3-x}...
- 1,405
4
votes
0
answers
462
views
Ramanujan's conjecture on modular forms and Riemann hypothesis
I just watched Kannan Soundararajan's talk on the distributions of valus of zeta and $L$-functions at virtual ICM 2022. In his talk, he introduced a theorem on Ramanujan's ternary form $\phi_{1}: x^{2}...
- 1,911
4
votes
1
answer
288
views
Selberg class definition and Riemann hypothesis
Looking at the Selberg class definition on Wikipedia, under "Comment on definition", there is this paragraph:
"The condition that the real part of $\mu_i$ be non-negative is because ...
- 1,121
4
votes
1
answer
387
views
Calculating the explicit constant – Siegel zeros and class numbers
Let $\chi$ denote the Legendre symbol of conductor $q$. A Siegel zero for the $ L $ series associated to $ \chi $, which we denote by $ L(s,\chi) $ is a real zero $ \sigma $ satisfying $ 1-\frac{c}{\...
- 537
3
votes
0
answers
178
views
Extending the analogy between cyclotomic units and elliptic units
There is a nice analogy between cyclotomic units and elliptic units given as follows:
Cyclotomic units are related to special values of the Riemann Zeta function. This is because the logarithmic ...
- 1,869
4
votes
0
answers
262
views
The link between Satake parameter and Godement-Jacquet L-function of an automorphic representation of $GL_{n}$
Origin of the question: I'm reading the following survey of K. Martin, more generally I'm looking for the "best way" to define L-function associated to an automorphic representation of a ...
- 1,105
8
votes
1
answer
388
views
Is $\frac{1}{L(1+it)}$ unbounded?
Let $\chi$ be a Dirichlet character and $L(s, \chi)$ be the corresponding L-function. Is $$\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 ...
4
votes
1
answer
237
views
Do Artin L functions have polynomial growth 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 ...
- 537
1
vote
0
answers
81
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,...
- 11
2
votes
0
answers
199
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 $...
- 21
1
vote
0
answers
187
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\...
- 1,223
3
votes
0
answers
234
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-...
- 6,932
2
votes
2
answers
241
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 $\...
- 617
3
votes
0
answers
83
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, ...
- 537
2
votes
1
answer
132
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 \...
- 1,869
1
vote
0
answers
116
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)/\...
- 1,869
9
votes
2
answers
569
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)]$, ...
user471019
8
votes
1
answer
304
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) ...
- 19.4k
1
vote
0
answers
116
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 ...
- 1,719
0
votes
1
answer
192
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 ...
user469908
4
votes
0
answers
99
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 ...
- 335
4
votes
0
answers
292
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 ...
- 6,932
5
votes
1
answer
2k
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 ...
- 6,932
6
votes
0
answers
227
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,\...
- 335
-3
votes
1
answer
198
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 ...
- 6,932
2
votes
0
answers
174
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 ...
- 313
1
vote
0
answers
111
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,...
- 875
3
votes
1
answer
111
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$$ for a cusp form $f(\tau)$ of weight-k, where $C$ is a path connecting $\tau =0$ and $\...
3
votes
0
answers
90
views
Study of relative class number of 'non-abelian' CM field by using L-functions
I'm currently interested in finding good upper bounds for the relative class numbers of non-abelian CM-fields.
So I'm looking for some references to learn the techniques that can be useful.
So far, I ...
- 1,043
1
vote
1
answer
302
views
Behaviour of a certain $L$ function at $s=1$
I was going through this paper. Corollary 7.3.4 says the $L$-function $L(s,\pi, \rm{sym}^4)$ is holomorphic except possibly at $s=0,1$ and gives a necessary and sufficient condition for it to have a ...
- 474
9
votes
0
answers
198
views
Unexpected patterns on the graph of an L-function on the critical line
Let $L(s)$ be the $L$-function associated to the (only) classical modular form of weight $26$ and level $1$.
The completed L-function $\Lambda(s)=2(2\pi)^{-s}\Gamma(s) L(s)$ is symmetric with respect ...
- 9,421
7
votes
0
answers
137
views
Are there natural Dirichlet series whose completions have poles in the region of absolute convergence?
The Selberg class of $L$-functions are Dirichlet series
$$ L(s, f) = \sum_{n \geq 1} \frac{a(n)}{n^s}, $$
satisfying certain properties that can be abbreviated as analyticity, a Ramanujan conjecture, ...
- 1,478
3
votes
1
answer
276
views
Watson's triple product for automorphic forms shifted by Maass rising operators
Let $\phi_i$ be a holomorphic Hecke eigencusp form of weight $k_i$ for $\Gamma = \mathrm{SL}_2(\mathbb{Z})$, or a Maass cusp form (we then say that $k_i=0$). We assume they are normalised such that $\...
- 143
-4
votes
1
answer
393
views
Scaled Riemann zeta function with no zero in the critical strip
Update: I added $exp[i\theta_k(s)]$ in the definition of $\eta^*(s)$ to address some critical convergence issues. Thanks for the contributors who pointed to these issues.
Prime numbers are denoted as $...
- 3,088
0
votes
0
answers
285
views
Is $\operatorname{Aut}(\mathcal{M})$ a fundamental group in Grothendieck's sense?
This question is a follow-up to Are there infinitely many L-rigs? and to Is an automorphic form of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ determined by its L-function?.
I copy paste a deepl ...
- 6,932
28
votes
1
answer
922
views
Relation between Schanuel's theorem and class number equation
(Crossposted on math stack exchange: https://math.stackexchange.com/questions/4040249/relation-between-schanuels-theorem-and-class-number-equation)
It was recently brought to my attention that there ...
- 595
9
votes
3
answers
579
views
Vinogradov-Korobov for Dirichlet L-functions?
Where can one find a Vinogradov-Korobov zero-free region for Dirichlet L-functions? It has to be in a standard reference, but I'm having a non-trivial time finding it.
- 19.4k
6
votes
2
answers
302
views
Functional equation and/or growth estimates for a shifted L function
Consider the $L$-series defined by
$$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 ...
- 19.4k
3
votes
2
answers
295
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. ...
- 9,421
8
votes
2
answers
314
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 ...
- 9,421