**35**

votes

**0**answers

863 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 ...

**13**

votes

**0**answers

781 views

### Special values of Artin L-functions

This question might be naive and might carry the heuristic that we are living in the best possible world a little too far. If so, I appreciate being told so.
Background: Stark's conjecture interprets ...

**10**

votes

**0**answers

250 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 ...

**10**

votes

**0**answers

382 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 ...

**10**

votes

**0**answers

498 views

### On the relation of special values of motivic L functions and partial zetas

Let $K$ be a number field, $L$ a finite abelian extension and $\chi \in \widehat{Gal(L/K)}$ a (non-trivial) character. If we multiply out the associated Artin L-function $L(\chi,s)$ we can write this ...

**10**

votes

**0**answers

914 views

### Are there analogues of Beilinson's conjectures for motives with coefficients?

There's a body of wisdom (following Beilinson, Bloch, Deligne, ...) relating mixed Tate motives, motivic cohomology, algebraic K-theory, special values of L-functions, and polylogarithms. My ...

**9**

votes

**0**answers

193 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

**0**answers

152 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 ...

**9**

votes

**0**answers

437 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!

**7**

votes

**0**answers

263 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 $\phi$ ...

**7**

votes

**0**answers

1k views

### What are “fractional motives”?

Kirti Joshi's musings mention "fractional motives". Do you know what are they good for and what the current state of constructions is for them?
Edit: Further cases of "fractional motives" as ...

**7**

votes

**0**answers

523 views

### Existence of multi-variable p-adic L-functions

What's the "state of the art" in constructing multi-variable p-adic L-functions for number fields?
More precisely: if K is a number field, and $K_{\infty} / K$ is an infinite Galois extension, ...

**5**

votes

**0**answers

226 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

**0**answers

127 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) = ...

**5**

votes

**0**answers

366 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\; ...

**4**

votes

**0**answers

55 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 ...

**4**

votes

**0**answers

100 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 ...

**4**

votes

**0**answers

193 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

86 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 ...

**3**

votes

**0**answers

93 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

**0**answers

135 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

**0**answers

121 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 ...

**3**

votes

**0**answers

182 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 ...

**3**

votes

**0**answers

345 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 ...

**2**

votes

**0**answers

110 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 ...

**2**

votes

**0**answers

103 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 ...

**2**

votes

**0**answers

111 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 ...

**2**

votes

**0**answers

218 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 ...

**2**

votes

**0**answers

255 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 ...

**2**

votes

**0**answers

191 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 ...

**2**

votes

**0**answers

98 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 ...

**1**

vote

**0**answers

168 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 ...

**1**

vote

**0**answers

251 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 ...

**1**

vote

**0**answers

60 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 ...

**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 ...

**1**

vote

**0**answers

78 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 ...

**1**

vote

**0**answers

321 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

126 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
$$
...

**0**

votes

**0**answers

133 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 ...

**0**

votes

**0**answers

48 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 ...

**0**

votes

**0**answers

75 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:
...

**0**

votes

**0**answers

84 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 ...

**0**

votes

**0**answers

112 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.

**0**

votes

**0**answers

96 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 ...

**0**

votes

**0**answers

87 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 ...

**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

**0**answers

252 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 ...

**0**

votes

**0**answers

265 views

### Does it exist a p-adic L function which interpolates the values of the complex one at positive integers?

I known that there are classical ways to construct $p$-adic $L$ functions for Dirichlet characters through $p$-adic integrals.
We fix a character $\chi$ modulo $Np^r$ with $N$ and $p$ coprime and an ...