is this map necessarily a field automorphism?

Question

Let $M$ denote the intersection of the Selberg class and the class of automorphic L-functions and let's define the automorphism group of $M$, denoted by $Aut(M)$, as the group (under composition) of bijections $\Phi$ from $M$ to itself such that $\forall(F,G)\in M^{2}$, $\Phi(F.G)=\Phi(F).\Phi(G)$ and $\Phi(F\otimes G)=\Phi(F)\otimes\Phi(G)$ where $\otimes$ is the Rankin-Selberg convolution. Let $deg$ be the map that maps an element of $M$ to its degree and $h$ the map $\Phi\in Aut(M)\mapsto\sigma_{\Phi}$ such that $deg\circ\Phi=\sigma_{\Phi}\circ deg$.
Is $\sigma_{\Phi}$ necessarily a field automorphism?
Thanks in advance.