**5**

votes

**1**answer

161 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

132 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

97 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

170 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

**1**answer

586 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

380 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

56 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

122 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

408 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

102 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

361 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

546 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 ...

**4**

votes

**0**answers

94 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

203 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

**0**answers

91 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

351 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

113 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

136 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

75 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

93 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

274 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

83 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

494 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

542 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

255 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

178 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

175 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

181 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

444 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

332 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

650 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 ...

**1**

vote

**0**answers

317 views

### P-Adic poly Bernoulli numbers

we can define p-adic Bernoulli polynomials by using q-integral on $Z_p$ and T.Kim's method, But how can we define p-adic poly-Bernoulli numbers and polynomials by using integral on $Z_p$ ?

**1**

vote

**0**answers

123 views

### Non-holomorphic L-functions

In some old notes, I found the following conjecture:
Let $\mathbb T$ denote the unit circle and let $\chi:{\mathbb N} \to {\mathbb T}$ be fully multiplicative. Then the L-series
$$
...

**9**

votes

**1**answer

574 views

### Montgomery's pair correlation function without RH?

In the theory of the Riemann zeta function, Montgomery's Pair correlation function is defined as
$$
F(\alpha) = \frac{1}{N(T)}
\sum_{T < \gamma, \gamma' < 2T} T^{i \alpha (\gamma - \gamma')} ...

**0**

votes

**0**answers

244 views

### Automorphism group of algebraic function fields

Let $K$ be a finite field and let $F/K$ be a function field. Is it possible to deduce the genus of $F/K$ from the automorphism group of $G=Aut(F/K)$?
Is it possible to do so if we know that $|G|$ is ...

**9**

votes

**0**answers

430 views

### Applications of Artin L-functions

Does anybody know a good reference which gives examples of applications of Artin L-functions to
"elementary" number theory? Many thanks!

**3**

votes

**4**answers

723 views

### Values of Dirichlet L-funcions at natural numbers

I want to know about reference of formulas for
$$
L(s,D)=\sum_{n=1}^\infty \left(\frac{D}{n}\right)\,n^{-s}
$$
for $s$ a positive integer number and $D$ a fundamental discriminant. For $s=1$ we have ...

**5**

votes

**1**answer

592 views

### The Correlation of the Mobius Function and Dirichlet Characters.

Let $\chi$ be a Dirichlet character, and define $\phi_\chi (n)$ so that it satisfies $$\sum_{n=1}^\infty \phi_\chi (n)n^{-s}=\frac{\zeta(s-1)}{L(s,\chi)}.$$
In other words
...

**1**

vote

**1**answer

426 views

### What are non-abelian $L$-functions?

I have heard people discussing the utility of $L$-functions, claiming that since they are essentially cohomological entities, they are "abelian" and therefore lack force.
From looking around on the ...

**16**

votes

**1**answer

685 views

### Distinct simple zeros of Dirichlet L-functions

Given a finite set of distinct primitive Dirichlet characters, $\chi_1, \dots, \chi_r$, is it known that the product of the L-functions, $$L(s):=\prod_{i=1}^r L(s,\chi_i),$$ has a simple zero? It's ...

**2**

votes

**0**answers

992 views

### Corvallis 1979 proceedings

These proceedings have long been freely available on the AMS website, but now it seems we can't even find them anymore (e.g. http://www.ams.org/publications/online-books/pspum331-index and ...

**15**

votes

**1**answer

1k views

### Two-variable p-adic L-functions of elliptic curves

Suppose $K$ is an imaginary quadratic field (with class number 1, for simplicity), $p \ne 2$ a prime split in $K$, and $K_\infty$ the $\mathbb{Z}_p^2$-extension of $K$.
If $E / \mathbb{Q}$ is an ...

**2**

votes

**1**answer

284 views

### How do you calculate the Euler factors of the imprimitive symmetric square at primes with bad reduction?

The reference for this question is Coates and Schmidt, Iwasawa theory for the symmetric square.
Let $G = \textrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}))$ and let $D_r \supseteq I_r$ be a ...

**16**

votes

**2**answers

1k views

### Computing (on a computer) the first few (non-trivial) zeros of the zeta function of a number field.

Let $M$ be the splitting field of
x^8 + 3*x^7 + 13*x^6 + 17*x^5 + 45*x^4 + 37*x^3 + 11*x^2 + 112*x + 108
over the rationals. If I've understood some tables ...

**1**

vote

**2**answers

649 views

### Functional equations relating to p-adic L-functions

Let f be a modular form of weight k for $\Gamma_0(N)$. Let us assume that $p\not\vert$N. Then we can construct 2 p-adic L-functions corresponding to the 2 roots $\alpha$ and $\beta$ of the equation ...

**3**

votes

**0**answers

340 views

### Non-vanishing of twists of L functions for GL(4)

Hello,
This is a question in the spirit of
Nonvanishing of central L-values of quadratic twists?
and the application I have in mind is to p-adic L-functions a la Ash-Ginzburg.
The question is ...

**13**

votes

**3**answers

1k views

### Nonvanishing of central L-values of quadratic twists?

Let $\pi$ be a cuspidal automorphic representation of GL(2) over a number field (if you want, assume it's $\mathbb Q$ and $\pi$ comes from a holomorphic modular form).
In the case $\pi$ has trivial ...

**10**

votes

**0**answers

378 views

### Is the Gouvea-Mazur problem related to symmetric square $L$-functions?

Here's an idea that I've found appealing but have never been able to get anywhere with.
One way to frame the Gouvea-Mazur question (for lack of a better term, since the original conjecture by the ...

**22**

votes

**1**answer

1k views

### Special values of L-functions as periods

If $M$ is a pure motive over $\mathbb{Q}$, one cas define its $L$-function $L(M,s)$ which conjecturaly is a meromorphic function over $\mathbb{C}$ with finitely many poles.
For example, when ...