Automorphisms of $\mathbb{C}$, Selberg class and surjectivity

Question

Let $F$ be an element of the Selberg class and $\sigma$ be a field automorphism of $\mathbb{C}$ such that $\sigma\circ F=F\circ\sigma$. Let $Fix_{\sigma}$ be the set of all complex numbers $z$ such that $\sigma(z)=z$ and $D_{F}$ be the domain of definition of $F$ (that is either the entire complex plane or $\mathbb{C}-\{1\}$). Is $F:Fix_{\sigma}\cap D_{F}\to Fix_{\sigma}$ surjective?
Thanks in advance.