Let $F$ and $G$ be two primitive functions of the Selberg class, and let $\mathbb{A}$ be the set of values taken by the function which maps a prime number $p$ to $a\_{p}(F)\overline{a\_{p}(G)}$. $\mathbb{A}$ is finite or countable, so that there exists $I\subset\mathbb{N}^\*$ such that $\displaystyle{\mathbb{A}=\bigcup\_{i\in I}\{A\_{i}\}}$, with $A\_i\neq A\_j$ whenever $i\neq j$.
Now, let $\delta$ be the function that maps $i\in I$ to $\displaystyle{\lim\_{x\to\infty}\frac{\#\{p\leq x, a\_{p}(F)\overline{a\_{p}(G)}=A\_i\}}{\pi(x)}}$.


Does Selberg's orthonormality conjecture imply $\displaystyle{\sum\_{i}\delta(i)A\_i=\delta\_{F,G}}$, where $\delta\_{F,G}=1$ if $F=G$ and $\delta\_{F,G}=0$ otherwise? If so, is such an implication in fact an equivalence?
Thank you in advance.



Selberg's orthonormality conjecture and density
No. (This is very unlikely to hold for anything except $L$-functions with only finitely many distinct coefficients; the simplest counter-example is to take an elliptic curve $E$ and its associated $L$-function $L(E,s)$, normalized to have critical line at $1/2$, and $F=G=L(E,s)$; for any non-zero real number $x>0$, due to the renormalization -- which by the way makes the question rather strange --, any prime $p$ for which $|a(p)|^2=x$, the Hasse-Weil coefficient $a\_p$ of the elliptic curve must satisfy $a\_p^2=px$; in particular $\sqrt{px}$ is an integer, and there are rather obviously only finitely many $p$ which can satisfy this for a given $x$. In other words, your function $\delta$ is identically zero, despite the fact that $\delta\_{F,F}=1$.)


One can also of course just take the $L$-function of the Ramanujan delta function, replacing just squareroots with powers $11/2$ in the right places.


