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
votes
2answers
121 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
224 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 ...
8
votes
0answers
118 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 ...
5
votes
0answers
185 views

A sum over zeros of L-functions in the paper “Chebychev's Bias”

Let $\varepsilon>0$ be small and \begin{align*} \widetilde{F}_{\varepsilon}(\xi):=\frac{4}{\varepsilon}\sum_{0<\gamma\leq \varepsilon^{-2}}\frac{\sin(\gamma \xi)\sin \frac{\gamma ...
5
votes
1answer
489 views

Main conjecture for elliptic curves

Suppose that $E$ is an elliptic curve defined over $\mathbb{Q}$, and that $p$ is a prime where $E$ has good ordinary reduction. Then one can define nonnegative integers $ \lambda_{E}^{alg} $, $ ...
28
votes
0answers
616 views

“Gross-Zagier” formulae outside of number theory

The Gross-Zagier formula and various variations of it form the starting point in most of the existing results towards the Birch and Swinnerton-Dyer conjecture. It relates the value at $1$ of the ...
2
votes
0answers
92 views

A $GL_1$ Voronoi formula

I want a functional equation for the function defined by the Dirichlet series, $$ D(s,a/q)= \sum_{n=1}^\infty \frac{e^{2\pi i n a/q}}{n^s}. $$ which sends $s$ to $1-s$ and preferably sends $a$ to ...
1
vote
0answers
36 views

Hasse-Weil L-Functions of CM Abelian Varieties

In Shimura's paper "On the Zeta Function of an Abelian Variety With Complex Multiplication", in his terminology, the `one-dimensional part' of the zeta function is identified with a Hecke $L$-function ...
1
vote
1answer
271 views

Order of some $L$-function at $s=0$

Sorry, I asked this two days ago, but this time I modified it to be easily read and added more specific explanation. I hope to get your illuminating comment on whether my approach is right. I am ...
2
votes
0answers
208 views

Convergence of certain L-series

Suppose $|a_{n}| \leq 1$ completely multiplicative function assuming real values. Suppose further that, $ L(s)=\sum_{n} \frac{a_{n}}{n^s} $ may be continued analytically to the left of $s=1$ a bit ...
4
votes
1answer
222 views

on the conductor of a scheme over $\mathbb{Z}$

Let $X$ be a regular, projective flat scheme over $Spec(\mathbb{Z})$, of relative dimension $d$, and look at the $L$-function $L(X, s)$, that is, the Hasse-Weil zeta function completed by the Gamma ...
2
votes
1answer
175 views

The effect of base change on the L-function of GL(2)?

Let $F$ be a local field (whose residue field is $q$) and $E$ its quadratic extension. Let $\pi$ be a irreducible principal series representation $\pi(\chi_1, \chi_2)$ of $GL_F(2)$ especially where ...
0
votes
0answers
94 views

reference needed: Dirichlet L-functions

Where is the logarithmic derivative at $s=0$ of the L-function of a Dirichlet character computed? Many people cite the Hurwitz formula but I was unable to find a suitable reference.
2
votes
2answers
538 views

L-functions and algebraic geometry

Robert Langlands commented in a letter to Deligne that perhaps some of the deepest problems of algebraic geometry lie in L-functions. I want to understand the general philosophy and the connection ...
0
votes
1answer
99 views

degree of an isobaric sum

I'm trying to understand a few things about automorphic L-functions. In page 5 of http://arxiv.org/pdf/1401.0390.pdf, the author mentions the isobaric sum decomposition ...
8
votes
1answer
337 views

Would Elliott-Halberstam conjecture follow from GRH?

The Wikipedia article about Elliott-Halberstam (EH for short) conjecture says that the so-called Bombieri-Vinogradov theorem, which is a weaker form of EH conjecture, is in some sense an averaged form ...
3
votes
0answers
128 views

Are quantities involved in Generalized Ramanujan Conjecture eigenvalues of some unitary operator?

If I'm not mistaken, every automorphic L-function $L(s,\pi)$ verifies $\displaystyle{L(s,\pi)=\prod_{p}L_{p}(s,\pi_{p})}$ where ...
1
vote
1answer
204 views

Elliptic units and Euler system

Maybe this question is quite obscure and ambiguous. I am really sorry for such ambiguity. My question is, what is the good thing we get from defining elliptic units and Euler system? There are lots ...
2
votes
0answers
220 views

Automorphisms of the L-function associated to an elliptic $\mathbb{Q}$-curve

Edited after Noam Elkies' comment: From what I understand (very few actually), there exist elliptic curves defined over some number fields $\mathbb{K}$ Galois over $\mathbb{Q}$ which are isogenous to ...
1
vote
0answers
160 views

Automorphicity of L-Factors of Zeta Functions

Associated to a variety over a number field $K$, one has a family of ``Hasse--Weil'' L-functions, which can be combined (as an alternating product) to give the Hasse--Weil zeta function of the ...
0
votes
0answers
197 views

Automorphisms of an L-function

Throughout this question, the term "L-function" will denote any element of the Selberg class. Following Strong automorphisms of the Selberg class, I define the group of automorphisms of an L-function ...
1
vote
1answer
160 views

Does $L(-1+it,f)\ll_f \log^c q(f)t$ hold ture?

Let $f$ be a holomorphic or Maass cusp form for $SL(2,Z)$. Define $L(s,f)=\sum_{n\ge 1}\frac{a_f(n)}{n^s}$, for $\Im s$ sufficiently large. Then $$L(-1+it,f)\ll_f \log^c q(f)t$$ holds, for some ...
9
votes
0answers
209 views

L-Functions of Varieties, Zeta Functions of Their Models

Let $k$ denote a number field, with algebraic closure $\bar{k}$. Take a smooth, projective variety $X$ over $k$. If $\mathfrak{p}$ is a prime of $k$, and $l$ is a rational prime different to the ...
2
votes
1answer
227 views

Does the property (P) holds true for the derivatives of $L$?

Let $$L(C,s)=\sum_{n=1}^\infty \frac{a_n}{n^s}$$ be the Dirichlet series of the Hasse--Weil L-function of an elliptic curve $C$ over $ℚ$. As $s$ takes on real negative values, there are trivial zeros ...
3
votes
1answer
298 views

On link between Riemann hypothesis and partial GRH

Is there a way to show that if the Riemann hypothesis holds for Dirichlet L-function associated to primitive Dirichlet character (excluding trivial character $\chi(1)$ which could be qualified of ...
1
vote
1answer
117 views

On properties of coefficients of Selberg Class L-function

The coefficient of Selberg Class L-function satisfy: $a_n <M_{\epsilon} n^{\epsilon}$ (for any $\epsilon >0$) and the $a_n$ are multiplicative. So I would like to know if it can be shown that ...
5
votes
1answer
155 views

Periods of Twists of Modular Forms

Let $f \in S_2(\Gamma_1(N))$ be an eigenform. By a theorem of Shimura, there are associated "periods" $\Omega_f^\pm$ such that, after normalizing by these periods, the L-function associated to $f$ ...
4
votes
1answer
126 views

Are supercuspidal reps of GL(2) uniquely determined by the rootnumber

Let $\pi, \pi'$ be a unitary, irreducible, supercuspidal representations of $GL_2(F)$. Does an equality of roots numbers $\epsilon(\pi, \psi, s) = \epsilon(\pi', \psi, s)$ for all $s \in \mathbb{C}$ ...
1
vote
1answer
93 views

Has a universality theorem been proved for the Davenport-Heilbronn L function?

The question is in the title: has a universality theorem in the sense of Voronin been proved for the Davenport-Heilbronn function, or do we expect such a theorem to hold true only for L functions that ...
0
votes
1answer
165 views

Poles of the L-series of the elliptic curve which has CM

Let $E/\mathbb{Q}$ be the elliptic curve which has CM. It is well known that the L-series of the elliptic curve which has CM is Hecke L-series. On the other hand, it is pointed out that Hecke ...
6
votes
1answer
539 views

Sufficient condition for Riemann Hypothesis?

Is there an L-function ($L_s=\sum_{n =1}^{\infty} \frac{a_n}{n^s}$) having a functional equation coming from a relation of the type [1]: $\sum_{n =1}^{\infty} a_n \; e^{-2\pi nx}= \frac{A}{x^k} ...
4
votes
1answer
306 views

subconvexity problem for $GL(3) × GL(2)$ $L$-function without involving in symmetric lift

A question in study of subconvexity topic puzzles me for a long time, which mabe a stupid question for many experts. I really wish someone to help me out, and any advice will be highly appreciated. ...
1
vote
0answers
51 views

Sign of the functional equation of L function and Shimura lift

I would like to know what happens to the root number of a half integral weight automorphic form (holomorphic or not), i.e. the sign of the functional equation of its L-function when we apply the ...
5
votes
0answers
119 views

What is the analogy between the Hilbert function and L-functions?

In his book Projective Varieties and Modular Forms, M. Eichler uses the notation $L(\lambda, M)$ for the Hilbert function of a finite graded $R=k[x_0, \dots, x_n]$-module $M$. So, $L(\lambda, M) = ...
2
votes
2answers
311 views

Summation of certain series

Suppose $f(n)$ is a periodic function with period $q$. Now from this paper we get that if $\displaystyle\sum_{n=1}^{q}f(n)=0$ then ...
3
votes
0answers
91 views

Do local L-functions/epsilon factors vary continuously with the Fell topology?

Edit due to the comment. Consider $G=GL(2)$ over a local field $F$. The Fell topology on the unitary dual of $G(F)$ is seperable. Given a sequence of irreducible unitary representations $(\pi_n)$ of ...
5
votes
0answers
349 views

a generalization of a formula of Shimura

Let $\phi$ be a $GL(2)$ automorphic form with Fourier coefficients $a(n)$ and $a(1)=1$. Obviously we have $L(s,\phi)=\sum \frac{a(n)}{n^s}$. Shimura have the following formula $L(s, Ad\; ...
7
votes
1answer
511 views

On Deligne's determinant of motives

This is a question about Deligne's conjecture on special values of L-functions. I have to confess that I've never understood the definition of the determinant which is supposed to give the right ...
3
votes
0answers
81 views

cluster variables and L-functions

There is something in common between cluster variables in the theory of cluster algebras, L-functions in number theory, namely the fact that both map direct sums to products, just like ...
3
votes
1answer
188 views

Extending the Shimura Lift to Non-Cuspidal Classical Modular Forms of Higher Level

The definition of the Shimura lift of a classical cusp form is well documented. Zagier and Kohnen define a modified version of the lift for a cusp form $g(z)=\sum a(n)q^n \in S_{k+1/2}^{+}(4)$ in the ...
1
vote
0answers
86 views

Theta lift to 1-dimensional vector space.

Hi! My question is very simple and it is about the theta lift of unitary group in global situation. Let $E/F$ be a quadratic number fields and $V,W$ be an $n$-dimensional and $1$-dimensional ...
8
votes
2answers
302 views

central/critical/special values of L-functions terminology

I have a question about the terminology for special values of L-functions. Is the following a correct description of standard usage: Suppose L(s) is an L-function which satisfies a functional ...
1
vote
0answers
107 views

Off critical line zeros for half integer weight $L$-functions

Let $f(z) = \sum_{n=1}^\infty A(n)n^{\frac{k-1}{2}}e(nz)$ be a modular form of weight $k$ for a half integer $k$. Put $$L(s,f) = \sum_{n=1}^\infty \frac{A(n)}{n^s} $$ to be the $L$-function. Further ...
2
votes
1answer
131 views

On the absolute convergence of the local-zeta integral.

Though I am in a situation considering only local-zeta integral, to explain my question briefly, let me ask it in quite general form. Let $f(s,g)$ be a two variable smooth good (in a suitable sense) ...
1
vote
0answers
70 views

Non vanishing and cuspidality of the theta lift of trivial representation.

Hi! Let E/F be a quadratic number field extension. Then we make some hermitian and skew hermition vector spaces and define unitary group on it.(namely U(1) and U(3)) Then, I am wondering whether the ...
0
votes
0answers
91 views

On the explicit formula of the height function occuring on the doubled Weil representation.

Hi! I am wondering the exact formula of height function of $GL(n)$ which occurs in the doubling Weil representation. To be more precise, let me introduce the basic setting for this. Let $F$ be the ...
1
vote
2answers
230 views

A question on twisted L-function

Hi! This is my first question here. In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character. For example, ...
0
votes
0answers
78 views

Can GRH for complex primitive Dirichlet characters fail with a single non-trivial zero off the critical line?

Can GRH for complex primitive Dirichlet character fail with a single non-trivial zero off the critical line? For real characters this is impossible because the non-trivial zeros are in ...
2
votes
2answers
453 views

About equivalent statements of the Birch and Swinnerton-Dyer Conjecture

The Birch and Swinnerton-Dyer Conjecture is well known in the current literature http://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture My question is about the possible equivalent ...
6
votes
1answer
513 views

A stupid question on automorphic l-function

This may be a silly question for experts in this area. But I am really suffering for not being able to compute local-L function of some automorphic representation. So, I post it hoping some ...