I was thinking of continuity since the map $f$ preserves the "inner product" defined for $(F,G)\in\mathcal{S}^{2}$ by $\displaystyle{\lim_{x\to\infty}\dfrac{1}{\log\log x}\sum_{p\leq x}\dfrac{a_{p}(F)\overline{a_{p}(G)}}{p}}$, so that $f$ appears to be some kind of an isometry of the Selberg class. And as generally speaking, an isometry is a continuous map...

But wouldn't it be possible to consider the conditions above as some kind of continuity assumptions that would make identity and complex conjugation the only possible maps such as condition 4) is true? A bit like the only continuous field automorphisms of $\mathbb{C}$ are namely the identity and the complex conjugation?

Could someone give a rigorous proof (always assuming Selberg's orthonormality conjecture) of the fact that there is no other automorphism than the identity and the complex conjugation? Thanks in advance.

Zen, I see no typo: can't you see the overline on $G$ ?

When I write "the same zeroes", I mean "the same zeroes with the same multiplicity for $\lambda_{F}$ and $\Lambda_{G}$", so that I don't think it's necessary to assume $F$ and $G$ are primitive.

Yes indeed, it is just the product of the functions $F_i$.

I'm not sure to understand your answer, but isn't the complex conjugation such an automorphism of $\mathcal{S}$ ?

Indeed the word "correspondence" may not fit exactly what I have on my mind, but my English is far from being perfect (I'm French). I didn't know that there were uncountably many functions in S, would you have some reference? By the way, still assuming Selberg's orthonormality conjecture, do both abelian varieties and Selberg's class form semi-simple categories? If so, the concept of functor from abelian varieties into Selberg's class, as you suggested, may be the good way to express my idea.