# Tagged Questions

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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### 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 ...
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### 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 $f$ ...
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### 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 $s \in \mathbb{C}$ ...
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### 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 ...
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### 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 L-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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 ...
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### 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, ...
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### 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 quadruples. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 http://www....
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### 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 ...
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### 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 ...
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### Consequences of the degree conjecture

the title is quite explicit: I would like to know the consequences of the degree conjecture for the Selberg class. Thank you in advance.
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 ...