Automorphisms of the Selberg class Hello,
assuming Selberg's orthonormality conjecture, let's consider bijective maps $f$ from Selberg's class $\mathcal{S}$ to itself such as:
1) $f$ maps a primitive function of $\mathcal{S}$ to a primitive function of $\mathcal{S}$,
2) $f$ maps a function of degree $d$ to a function of degree $d$,
3) for every $F$ in $\mathcal{S}$, $n_{f(F)}=n_{F}$, where $n_{F}$ is the integer involved in Selberg's conjecture $A$,
4) if $F=F_{1}^{e_{1}}.F_{2}^{e_{2}}....F_{k}^{e_{k}}$, then $f(F)=f(F_{1})^{e_{1}}.f(F_{2})^{e_{2}}....f(F_{k})^{e_{k}}$ (so that $f$ is "strongly multiplicative"),
5) if $f$ verifies the above conditions, then so does the inverse of $f$.
The set of all such maps makes a group for the composition law. I would like to know whether this group is isomorphic to the absolute Galois group of the field of the rational numbers or not.
Thanks in advance.
 A: Most elements in $\mathcal{S}$ probably don't have algebraic Dirichlet coefficients (even after scaling), e.g. this is conjectured for the $L$-functions of non-CM Maass forms. So I don't see that the Galois group in question acts on $\mathcal{S}$. I think it is fair to conjecture that the only automorphism of $\mathcal{S}$ is the identity.
A: Doesn't twisting by (finite-order) Dirichlet characters satisfy all the properties you listed?
A: The group in question really seems to be the direct product, over all $d$, of the full symmetric group of the set of primitive functions of degree $d$, extended by multiplicativity (condition 2 is automatic from the others because the integer $n_F$ is supposed to be the sum of squares of the multiplicities of the primitive factors). So this is a seriously huge group.
(About the last comment: the group can not act transitively, since it is supposed to preserve the degree; but the orbits are the same as the possible degrees).
A: In fact, I have quite good reasons to think that the above conjecture (the only automorphisms of $\mathcal{S}$ are the identity and the complex conjugation) is equivalent to the Riemann Hypothesis for the whole Selberg class. Certainly one would wish for a stricter proof here... :-)
EDIT: Considering David Hansen's answer, it appears that the group of automorphisms of $\mathcal{S}$ might be richer than I first expected. In fact I may have been a bit hasty saying "is equivalent to". An interesting question would be to ask whether the action of this group on $\mathcal{S}$ is transitive or not.
