Finiteness of the extension of $\mathbb Q$ generated by Frobenius traces, in finite characteristic

Let $\mathscr{F}_0$ be a $\bar{\mathbb{Q}}_l$-sheaf on a scheme $Z_0$ of finite type over $\mathbb{F}_q$. We show the existence of a finite type extension $E\subset\bar{\mathbb{Q}}_l$ of $\mathbb{Q}$ such that all local factors of the $L$-function of $\mathscr{F}_0$ have coefficients in $E$. When $Z_0$ is normal and connected, and $\mathscr{F}_0$ is an irreducible $l$-adic local system whose determinant is of finite order, $E$ can be taken to be a finite extension of $\mathbb{Q}$.