Hello everyone,

I would like to know whether, assuming Selberg's orthonormality conjecture, it would be possible to establish a "natural" correspondence between abelian varieties and functions belonging to the Selberg class S in such a way:

1. One associates to a simple abelian variety a primitive function of S,
2. One associates to an abelian variety of dimension d a function of S of degree d,
3. If V is an abelian variety isogenous to a product of abelian varieties of lower dimensions V_1, V_2, ... V_n, then the function F of S related to V is the product of the F_i where F_i is the function of S related to V_i.

Thank you in advance.