Hello,

this question is a follow-up from Structure groups and a special class of L-functions

Let $\tilde{\phi}:\tilde{\mathcal{S}}\to\tilde{\mathcal{S}}$ be an automorphism of $\tilde{\mathcal{S}}$, and let's say that $\tilde{\phi}$ is isometric if there exists an affine isometry $\phi:\mathbb{C}\to\mathbb{C}$ such that for all $F\in\tilde{\mathcal{S}}$ and for all $s\in\mathbb{C},\ \ s\neq 1$, $(\tilde{\phi}(F))(s)=(\phi\circ F \circ \phi^{-1})(s)$.

Is it true that all automorphisms of $\tilde{\mathcal{S}}$ which restrict to identity on $\tilde{\mathcal{S}}_{\mathbb{R}}$ are isometric?

Thanks in advance.