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 Lfunctions. 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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