ag.algebraic geometry - Behaviour of Zeta-function under Finite Morphism

You should be looking not at just zeta functions, but at the L-functions.

Then yes, for a finite etale Galois morphism the identity should be

        Z(Y) = Z(X) * L(X, pi_1) * L(X, pi_2) * ...

(where the product is over summands of the regular representation of Galois group of the morphism, Z being the L-function of the trivial representation.) This is in no way restricted to finite fields — in fact the idea as well as the notation comes from theorem about Dirichlet L-functions.

The proof is that by a definition of what is L-function it can be written either for a trivial mixed sheaf on Y (LHS) or for its pushforward on X (RHS).