**1**

vote

**1**answer

120 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

**1**answer

307 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

**0**answers

219 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

**1**answer

229 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

**1**answer

211 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

**0**answers

115 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

**2**answers

611 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

**1**answer

138 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

**1**answer

459 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

**0**answers

137 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 ...

**3**

votes

**1**answer

251 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

**0**answers

199 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 ...

**1**

vote

**1**answer

192 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 ...

**10**

votes

**0**answers

263 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 ...

**1**

vote

**1**answer

670 views

### Backlund counting formula for Dirichlet L-functions? [closed]

Are there published works on the analog of Backlund's counting formula for Riemann zeros on the strip involving Riemann-Siegel theta, but for Dirichlet L-functions? We found papers with the analog ...

**1**

vote

**0**answers

254 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

**1**answer

327 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

**1**answer

123 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

**1**answer

178 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

**1**answer

138 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

**1**answer

100 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

**1**answer

180 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 ...

**5**

votes

**1**answer

636 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

**1**answer

468 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

**0**answers

74 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

**0**answers

130 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

**2**answers

417 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

**0**answers

126 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

**0**answers

367 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

**1**answer

570 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 ...

**6**

votes

**0**answers

137 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

**1**answer

221 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 ...

**2**

votes

**0**answers

104 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

**2**answers

414 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

**0**answers

138 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

**1**answer

139 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

**0**answers

80 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

**0**answers

97 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 ...

**2**

votes

**2**answers

321 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

**0**answers

89 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 ...

**3**

votes

**2**answers

507 views

### About equivalent statements of the Birch and Swinnerton-Dyer Conjecture [closed]

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

**1**answer

575 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 ...

**8**

votes

**2**answers

262 views

### Is the following the right definition of $L$-functions (on the Galois side)?

This question may be too elementary for this forum, but I have asked it on math stackexchange and didn't get an answer. I have now deleted it so there wouldn't be duplicates... Here is the question as ...

**6**

votes

**1**answer

1k views

### derivatives of Artin L-functions

This is a vague question: I'm sorry for that.
Let's start with $\chi$ a (primitive odd) Dirichlet character modulo $n$ and look at the corresponding L-function
$$
L(s, \chi)=\sum ...

**4**

votes

**0**answers

199 views

### Do infinite and ramified local factors of the Dedekind zeta function of a tame number field characterize its local root numbers?

Let say you have two number fields, that are tamely ramified, and suppose that the $p$-part of their Dedekind zeta functions coincide for all prime $p$ which is ramified in either field. Suppose ...

**3**

votes

**0**answers

187 views

### Is this extension of the Selberg class trivial?

I came across the following modification of the Selberg class in some of my work (see below), and while I've moved on in some sense -- I submitted the paper in question -- I can't get it off of my ...

**1**

vote

**1**answer

185 views

### Understanding the determinant of the action of Frobenius on the character group of the toric part of the reduction of the Jacobian of a curve.

I am trying to understand a certain sentence in a paper that I am reading. Let me start with some notation/background. (For a motivation of why this should be interesting, see below, under the ...

**4**

votes

**2**answers

478 views

### Blueprint of L-functions and need for introducing them ( Hasse-Weil L-functions )

Dear All,
This question may appear elementary to all the experts in number theory , but forgive me. I really wanted to know how did the $L$-functions came into existence, especially the Hasse-Weil ...

**0**

votes

**0**answers

335 views

### Linking L function dynamics with behavior close to s = 1 ?

A division, found on a sample set of semi-stable elliptic curves, calls for interpretation regarding the Birch and Swinnerton-Dyer conjecture and the dynamic behavior of the L functions involved.
...

**0**

votes

**1**answer

669 views

### Interplay between Riemann and Swinnerton-Dyer

Hello everyone,
After reading RH ( Riemann's Hypothesis ) and Swinnerton-Dyer conjecture, I asked myself why can't RH hold for $L$-Functions ( Hasse-Weil L-function ). In particular the GRH imposes ...