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, ...

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3
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1answer
100 views

gamma-factor of a primitive element of the Selberg class

Suppose $F$ is a primitive element of the Selberg class and $\displaystyle{\prod_{j=1}^{r}\Gamma(\lambda_{j}s+\mu_{j})}$ with $r>1$ the product of Gamma functions appearing in the gamma factor ...
0
votes
0answers
41 views

Automorphism group of the gamma factor of a certain type of L-function [on hold]

Let $F$ be an element of the Selberg class with polynomial Euler product, $\gamma_F$ its gamma factor appearing in the functional equation of $F$, which is defined up to a multiplicative factor. Is ...
0
votes
1answer
175 views

Does unique factorization for automorphic L-functions imply a weakened form of Ramanujan conjecture?

Selberg orthonormality conjecture for automorphic L-functions was proven under Ramanujan conjecture, and SOC itself implies unique factorization for those L-functions. My question is: does the ...
3
votes
1answer
104 views

What is the analytic conductor of this Hecke L-function?

Following Iwaniek and Kowalski, S5.10, page 130 we consider an angle character $\xi_k$ on the Gaussian integers $\mathbb Z[i]$ defined by $ \xi_k(\mathfrak a) = \left(\frac{\alpha}{|\alpha|}\right)^k ...
0
votes
0answers
46 views

Function series involving a suite of imprimitive Dirichlet characters and a zero of $L(\chi,s)$

I have difficulties to understand the behavior of following suite of functions near zero : $$F_P(x)= \sum\limits_{n=P}^{\infty} \chi_P(n) f(nx)$$ With $f(x) = x^{-s_0} e^{-x}$ where $s_0$ is a non ...
4
votes
3answers
312 views

what is exactly the difference between the Selberg class and the set of Artin L-functions?

The question is in the title: from what I read in the answer to another question, Artin L-functions are conjecturally cuspidal automorphic L-functions for some algebraic group that can be transfered ...
3
votes
1answer
139 views

Subconvexity bound for Hecke $L$-functions in the $s$-aspect

Let $L(s,\chi)$ be the $L$-function of a non-trivial Hecke character of a general number field $K$, so that $L(s,\chi)$ which has no pole or zero at $s=1$. I am looking for a reference for upper ...
8
votes
1answer
240 views

Primer on Eisenstein series

My apologies if this question is a duplicate. I seached, and the closest I could locate is this question, which has very intriguing and intractable (for me) responses. In my continuing journey of ...
3
votes
0answers
292 views

Do those manifolds atrached to L-functions give rise naturally to motives? [closed]

Edited after Will Sawin's comment: Consider the set $\mathcal{M}$ of all automorphic L-functions belonging to the Selberg class. Such a set is closed for the product $.$ and the tensor product ...
8
votes
1answer
366 views

Is this theorem on $L$-functions known?

Notations For $f$ a meromorphic function on a domain $\Omega\subseteq \textbf{C}$, we shall say for convenience that $f$ is represented by an Ordinary Dirichlet Series (ODS) if $f$ can be written ...
5
votes
1answer
120 views

Bounding a Sum of Adjoint L-Function Values

Fix integers $k\geq2$ and $N>1$, and let $S(k,N)$ denote the normalized new Hecke eigenforms in $S_k(\Gamma_1(N))$. [If it makes my question easier to answer, feel free to replace this with ...
0
votes
0answers
113 views

seminar about the strong multiplicity one for the Selberg class

Very recently, a seminar took place in Seoul with Haseo Ki as an invited speaker to talk about the strong multiplicity one theorem for the whole Selberg class that he did manage to prove. I would like ...
8
votes
2answers
384 views

Averages over integer points of the sphere

A paper of William Duke proves that integer points on the sphere are equidistributed: $$ V_n = \{ (x,y,z) \in \mathbb{Z}^2 : x^2 + y^2 + z^2 = n \}. $$ Up to reflections across the $x$, $y$ and $z$ ...
1
vote
0answers
105 views

dimension of a scheme and degree of an L-function [closed]

I try to understand correctly the notion of scheme, as Serre in the second volume of his Oeuvres defines zeta and L-functions in this context. What seems interesting to me is that he states a theorem ...
5
votes
1answer
188 views

Symmetry type of non-cohomological automorphic forms

By Katz-Sarnak philosophy a family of $L$-functions would have a symmetry type which would reflect the statistics of $L$-functions, such as low lying zeros and moments. Shin-Templier's paper on ...
4
votes
1answer
233 views

Irrationality of Dedekind zeta values

For Riemann's zeta function, one knows that: $\zeta(2n)$ is irrational (because a rational multiple of $\pi^{2n}$ is) $\zeta(3)$ is irrational (proved by Apéry) and a few other results like "there ...
4
votes
2answers
308 views

References for general Hasse-Weil zeta function

Most research on the Hasse-Weil zeta function focuses on some particular type of algebraic variety, and general surveys usually deal mostly with the better understood elliptic curve case. I am ...
0
votes
0answers
145 views

Does the proportion of non trivial zeroes of given real part increase on [0,1/2] for all L-functions?

Here what I call an L-function is either an element of the Selberg class or an automorphic L-function. For such a function $F$, $x\in [0,1/2]$ and $T\gt 0$, let's define $\delta_{F,T}(x)$ such that ...
7
votes
1answer
437 views

Tate's thesis for Artin L-functions

As far as I know, Tate's thesis has been successfully applied in two fronts: Hecke L-functions, by Tate and Iwasawa (and Teichmüller, Witt, Schmid) Automorphic L-functions, by Jacquet, Shalika, ...
27
votes
3answers
1k views

Underlying idea for (automorphic) L-function?

Edit: So with a few more months of math under my belt, I recognize some of the issues with this question. I still hope for an answer, so let me say a few things. Within the Langlands philosophy, ...
2
votes
0answers
137 views

Functoriality for non-split orthogonal groups

I am trying to understand the functoriality conjectures of Langlands. We know that the functoriality conjectures imply that automorphic $L$-functions of a connected reductive group are equal to ...
3
votes
0answers
91 views

Oscillatory integral moments of $L(\frac{1}{2} + it, f \times f)$

Understanding moments and subconvexity bounds for $L$-functions is a big topic with a lot of activity. I'm currently looking at a related problem, bounding $$ \int_0^T L\left(\tfrac{1}{2} + it, f ...
0
votes
0answers
52 views

Arguments of Dirichlet coefficients of prime index of primitive elements of the Selberg class

Let $F$ and $G$ be two primitive elements of the Selberg class such that for $\Re(s)>1$, $\displaystyle{F(s)=\sum_{n>0}\dfrac{a_{F}(n)}{n^s}}$ and ...
7
votes
0answers
288 views

Lindelof Hypothesis implying Selberg Eigenvalue Conjecture?

(General) Lindelof Hypothesis which says for any $L$-function we have $$L(1/2+it)\ll Q(t)^{\epsilon}$$ for any $\epsilon>0$ where $Q(t)$ is the conductor of $L(s)$ at $t$. For a Maass form $\phi$ ...
3
votes
1answer
173 views

Tensor product of two elements of the Selberg class

Maybe too easy a question for most members of this site, but suppose whenever $F$ and $G$ belong to the Selberg class, then so does $F\otimes G$ where the considered tensor product of $F$ and $G$ is ...
3
votes
1answer
138 views

Lower bound of first moment of $L$-function on $\mathrm{GL}(3)$

Let $\pi$ be an automorphic form on $\mathrm{GL}(3,\mathbb{A}_{\mathbb{Q}})$. Do we know any case that \[\int_0^{T} \left|L(\frac{1}{2} + it, \pi)\right| dt \gg T\] holds unconditionally? I know the ...
0
votes
0answers
81 views

Are the first ten zeros of this Dedekind zeta function non-simple?

This question asks about the zeros of the zeta function of the number field with defining polynomial: ...
1
vote
0answers
183 views

on the Rankin-Selberg L-function

Let $n,m$ be two different positive integers. I heard that for cuspidal tempered automorphic representations $\pi_{n}$ and $\pi_m$ of $GL_n$ and $GL_m$, the Rankin-Selberg L-function $L(s,\pi_n ...
2
votes
1answer
132 views

convergence of L-functions of curves

Let $C$ be a smooth projective curve over $\mathbb{Q}$. Its associated L-function is defined by $$ L(C, s)=\prod_{p \text{ prime}} L_p(C, s), $$ where, if $p$ is a prime of good reduction, ...
5
votes
1answer
146 views

p-adic L-function of curves

Given a smooth projective curve $C$ over $\mathbb{Q}$ one has the $L$-function $L(C, s)$ and the Beilinson conjectures predict its values at integers $s=n$ in terms of regulators. Is there a p-adic ...
4
votes
0answers
60 views

Archimedean $\varepsilon$-factors

Let $K$ be either $\bf R$ or $\bf C$. Let $p$ and $q$ be integers with $p \leq -1$, $q \geq 0$, and $p+q=-1$. Consider the Hodge structure $M = M(p,q)$ over $K$ with coefficients in $\bf R$, defined ...
1
vote
1answer
132 views

On the pole of local L-function

Let $F$ be a number field and $v$ a finite place of $F$. Let $\chi_v$ be a unramified unitary character of $F_v$. Then we define local L-function $L_v(s,\chi_v):=\frac{1}{1-\chi_v(\omega)q^{-s}}$ ...
4
votes
1answer
157 views

Characterizing the newforms s.t. the associated symmetric square $L$-function has a pole

I have a straightforward question. Let $f$ be a holomorphic cusp form of weight $k$, level $N$, and nebentypus $\chi$ that is new in the sense of Atkin-Lehner theory. Write its Fourier expansion at ...
1
vote
1answer
318 views

Reference request: Grothendieck´s period conjecture?

I would like to know if Grothendieck published something about this conjecture? Is there some book (or expository article) about this conjecture? Is there any connection between this conjecture and ...
6
votes
1answer
313 views

Motivic L-function vs motivic zeta function

Let $M$ be a pure motive over a field $k$. Roughly speaking, the L-function of $M$ is the product over all primes $p$ of $$L_p(M,s)=\det(I-Fr_p|_{M_\ell^I} N(p)^{-s})^{-1}$$ where $Fr_p$ is a ...
8
votes
0answers
214 views

Symmetric Fifth Power Lift of GL(2) Automorphic Form

Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, ...
9
votes
1answer
280 views

Is the adjoint L-function on GL(m) holomorphic?

Let $\pi$ be an automorphic representation on GL($m$)/$\mathbb{Q}$. Define $$L(s,\pi,Ad):=\frac{L(s,\pi\times\overline{\pi})}{\zeta(s)}.$$ This is an L-function with euler products of degree $m^2-1$. ...
3
votes
2answers
426 views

Local factors of Hasse-Weil zeta function - what do they have in common?

Let $X$ be a regular scheme, flat and of finite type over $Spec(\mathbb{Z})$ (add "projective" if you want). Then the Hasse-Weil zeta function of $X$ is defined as a product over all prime numbers of ...
5
votes
1answer
235 views

absolute convergence of Rankin-Selberg series

Let $\pi$ and $\pi'$ be two general automorphic representation on $GL(n)$ and $GL(n')$ over $\mathbb{Q}$. I heard that the rankin-selberg convolution L-function $L(s,\pi\times\pi')$ is absolutely ...
-3
votes
1answer
385 views

Is SOC known to imply the Grand Riemann Hypothesis? [closed]

I'm currently working on a conditional proof of the Grand Riemann Hypothesis, which is based on the assumption that every field automorphism of $\mathbb{C}$ that commutes with an element of the ...
2
votes
1answer
111 views

Are there L-functions of degree 1 that aren't Hecke L-functions?

I don't know of any examples and I don't know of any results which prohibit them
2
votes
1answer
459 views

GL(2) Local Langlands and Artin's L-function

The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome. In light the local Langlands correspondence, what ...
2
votes
1answer
255 views

Automorphic L-functions over $GL_n( \mathbb{Q} )$

A paper by Kaczorowski & Perelli arXiv:1207.2312 dealing with the elements of the Selber class with degree two suggests that $S_2$ coincides with the automorphic l-functions over $GL_2( \mathbb{Q} ...
3
votes
0answers
104 views

Functoriality for triple product GL(2) x GL(2) x GL(2)

Let $f$, $g$ and $h$ be three general automorphic forms on GL(2). Do we know that $L(s, f\times g\times h)$ comes from an automorphic form on GL(8)?
3
votes
2answers
162 views

Main conjecture for elliptic curves invariant under a $\mathbb{Q}$-isogeny

Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ \mu_{E}^{alg} $, $ ...
5
votes
1answer
285 views

p-adic L-functions of modular forms: why the condition $v_p(\alpha)<k-1$?

Let $f$ be a modular form (cuspidal, new, eigenform) of weight $k$ and level $N$. Let $p$ be a prime number not dividing $N$. In order to construct a $p$-adic $L$-function $L_p(f, s)$ interpolating ...
0
votes
0answers
89 views

Has universality been definitely established for the whole Selberg class?

I juste googled to get some insight about universality for l-functions belonging to the Selberg class, but it seems that the proof requires the validity of the prime number theorem for the considered ...
3
votes
2answers
232 views

l-functions of calabi-yau varieties

This question might not be suitable for MO since i know nothing about Calabi-yau varieties aside the fact that they are used in string theory to compactify additional dimensions, but still, it makes ...
6
votes
1answer
438 views

Generalization of Watson's triple product

In Watson's thesis (page 51) we can find his beautiful triple product formula. My question is that does there exist a generalization of this formula? By generalization, I mean: If $\phi_n$'s are ...
10
votes
0answers
169 views

Hodge Decompositions and Gamma Factors of Hasse--Weil L-Functions

Let $X$ be a projective variety over a number field $K$. If $\mathfrak{p}\unlhd\mathcal{O}_K$ is a prime ideal we can regard the Euler factor of its $m$th Hasse-Weil $L$-Function at $\mathfrak{p}$ as ...