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.