# 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 $\sigma$ such that $\langle F_i, F_j\rangle=\langle F_{\sigma(i)}, F_{\sigma(j)}\rangle$ where $\langle F, G\rangle=\lim_{x\to\infty}\dfrac{1}{\log\log x}\sum_{p\leq x}\dfrac{a_{p}(F)\overline{a_p(G)}}{p}$ and $a_p(F)$ (resp. $a_p(G)$) being the $p$-th coefficient of the coefficient of the Dirichlet series defining $F(s)$ (resp. $G(s)$) for $\Re(s)>1$?
Thanks in advance.

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Permutations of what? –  Greg Martin Mar 29 '13 at 18:24
Sorry, permutations of the set of the positive integers. –  Sylvain JULIEN Mar 29 '13 at 18:56
I do not believe that the number of primitive elements of the Selberg class is countable. For instance, $F(s)=L(s+ia,\chi)$ is primitive when $\chi$ is a primitive Dirichlet character and $a \in \mathbb{R}$. –  Micah Milinovich Mar 29 '13 at 19:52
The formulation of your question is strange : the condition you want depends only on the values of $\sigma$ at $i$ and $j$ -- do you realy fix $i$ and $j$ or do you allow them to vary? –  François Brunault Mar 29 '13 at 21:16
@Micah: following mat.unimi.it/users/molteni/research/papers-pdf/…, I consider that two primitive elements of the Selberg class are equivalent if the second is the twist of the first. –  Sylvain JULIEN Mar 29 '13 at 21:18
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