Questions about Selberg class and the related conjectures such as the analogue of Riemann Hypothesis, Selberg's orthonormality conjecture, degree conjecture, general converse conjecture that says the Selberg class exactly consists of automorphic L-functions, etc.

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Galois classes of L-functions

Around one month ago, I posted on math.stackexchange a draft I wrote in which I define the notion of Galois class of L-functions: see ...
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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 ...
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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 ...
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Strong automorphisms of the Selberg class

Following Automorphisms of the Selberg class, I define strong automorphisms of the Selberg class by adding as an hypothesis that every invariant of $F$ (i.e all the $H$-invariants, conductor and root ...
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Automorphisms of $\mathbb{C}$, Selberg class and surjectivity

Let $F$ be an element of the Selberg class and $\sigma$ be a field automorphism of $\mathbb{C}$ such that $\sigma\circ F=F\circ\sigma$. Let $Fix_{\sigma}$ be the set of all complex numbers $z$ such ...
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Would countability conjecture and degree conjecture imply unique factorization for the Selberg class?

I already asked this question on MSE but didn't get any comment or answer, so I ask it here. Assuming countability conjecture as stated in ...
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A possible application of representation theory to Galois classes of L-functions

I define the notion of a Galois class of L-function as follows: $A$ is a Galois class of L-functions if and only if the following conditions simultaneously hold true: 1) $A$ is a subset of the ...
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Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC?

Would a proof of both (G)RH and Montgomery's pair correlation conjecture imply SOC? It seems, judging by the abstract of a 2002 paper of Ram Murty and a possibly Romanian co-author published on ...
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Is a certain group related to a primitive L function isomorphic to $Gal(\overline{\mathbb{Q}}_{\ell}/\mathbb{Q}_{\ell})$ for some $\ell$?

I define the notion of "Galois class of L functions" in the following way: $A$ is a Galois class of L functions if and only if the follwing three conditions hold simultaneously: 1) every element ...
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Automorphisms of S and representations

EDIT July 22nd 2013: I add further details in bolded sentences: Assuming Selberg's orthonormality conjecture and following Automorphisms of the Selberg class, I define automorphisms of the Selberg ...
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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 ...
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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 ...
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Invariants of an L-function

For any element $F$ of the Selberg class, one can define invariants of $F$ such as the degree, the conductor, the h- invariants, etc. My question is: these any known set of invariants such that a ...
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Selberg's orthonormality conjecture and permutations

Let $(F_n)_{n>0}$ be an enumeration of the primitive elements of the Selberg class. Let $F_i$ and $F_j$ be such primitive L-functions. What is known about the group $G(i,j)$ of permutations ...