**5**

votes

**0**answers

103 views

### Moments of completed L-functions?

This is a follow up question to this one.
It seems that results on moments of L-functions, that is, estimates for integrals of the form
$$\int^{T}_1|\zeta(\sigma+it)|^{2k}dt$$
are typically for the ...

**5**

votes

**2**answers

387 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 ...

**22**

votes

**1**answer

2k views

### Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...

**2**

votes

**1**answer

123 views

### Which values of symmetric square $L$-functions are critical?

I've recently been learning about the special values of symmetric square $L$-functions of modular forms.
If $f$ is a cuspidal modular eigenform (of some weight $k \ge 2$) then its symmetric square ...

**2**

votes

**0**answers

127 views

### $\frac{1}{2}<\sigma<1$, is $f(n) = \Bigl| \,1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|$ from $O(\log n)$?

We have $\frac{1}{2} < \sigma < 1$ and
$$
f(n) = \Bigl|\, 1+ \frac{1}{2^{\sigma + i n}} + \cdots + \frac{1}{n^{\sigma + i n}} \Bigr|
$$
. My goal is proving this statement that $|f(n)|$ is ...

**7**

votes

**1**answer

203 views

### Numerically double-checking formula with L-values

I'm working with a special case of Ichino's triple product formula, which for classical holomorphic newforms $f$, $g$ ,$h$ of weights $k$, $m-k$, $m$ (and central characters satisfying $\chi_f \chi_g ...

**1**

vote

**1**answer

118 views

### Question about mean square estimate for sums of Dirichlet coefficients of Symmetric Power $L$-functions

I have a question related to Coefficients of Symmetric power $L$-functions and I would be grateful if you could answer it.
Let $\lambda_{Sym^rf}(n)$ be the $n$th Dirichlet coefficient of ...

**44**

votes

**2**answers

2k 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 ...

**8**

votes

**0**answers

165 views

### Modular factorization of Dedekind zeta functions

It is well known that for abelian number fields, the factorization of its Dedekind zeta function goes like this:
$$\zeta_K=\zeta\prod_\chi L(s,\chi)$$
with the Dirichlet characters distinct and ...

**10**

votes

**1**answer

390 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 ...

**11**

votes

**2**answers

458 views

### Easiest way to see that $\zeta_{\mathbb{Z}[i]}(s) = \zeta(s) L(s, \chi)$?

As the question suggests, what is the easiest way to see that$$\zeta_{\mathbb{Z}[i]}(s) = \zeta(s)L(s, \chi)?$$Here, $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to \mathbb{C}^\times$ ...

**5**

votes

**1**answer

166 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 ...

**5**

votes

**0**answers

140 views

### Explicit bounds for exceptional zeros and/or $L(1,\chi)$ for real $\chi$

I would like to have an explicit upper bound (that is, one with explicit constants) for a possible real zero $\beta$ for $L(1,\chi)$ for real Dirichlet characters $\chi$. I need such a bound for real ...

**4**

votes

**1**answer

87 views

### Analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$, does $L_c(s, \chi)$ necessarily equal $1$?

Consider an analogue of Dirichlet $L$-function for $\mathbb{F}_q[T]$. Let $g \in \mathbb{F}_q[T]$, $g \neq 0$, let $\chi: (\mathbb{F}_q[T]/(g))^\times \to \mathbb{C}^\times$ be a homomorphism, let $c ...

**3**

votes

**1**answer

129 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

**1**answer

186 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 ...

**10**

votes

**2**answers

309 views

### reference help needed on a fact about poles of L-functions

Suppose $\pi$ and $\rho$ are cuspidal automorphic representations on $GL(n)$ and $GL(m)$ respectively. Then the L-function $L(s,\pi \times \rho)$ has a pole iff and $m=n$ and $\pi$ is isomorphic to ...

**27**

votes

**3**answers

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, ...

**3**

votes

**1**answer

118 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 ...

**37**

votes

**15**answers

5k views

### Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$?
Background: The non-vanishing ...

**0**

votes

**0**answers

55 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

**3**answers

323 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 ...

**6**

votes

**0**answers

142 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

146 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 ...

**1**

vote

**1**answer

683 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 ...

**2**

votes

**0**answers

302 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

**1**answer

390 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 ...

**0**

votes

**0**answers

118 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 ...

**3**

votes

**1**answer

177 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 ...

**5**

votes

**1**answer

194 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 ...

**1**

vote

**0**answers

79 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

139 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 ...

**8**

votes

**2**answers

404 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 $z$ ...

**4**

votes

**1**answer

252 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 ...

**1**

vote

**1**answer

133 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 ...

**4**

votes

**1**answer

1k 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 ...

**7**

votes

**1**answer

463 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, ...

**2**

votes

**0**answers

151 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

**0**answers

92 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

**0**answers

54 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

**0**answers

310 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 $\phi$ ...

**6**

votes

**1**answer

462 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 ...

**3**

votes

**1**answer

144 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

**0**answers

83 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:
...

**2**

votes

**0**answers

202 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

**1**answer

135 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

**1**answer

147 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

**0**answers

67 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

**1**answer

138 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

**1**answer

165 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 ...