Let $X_0$ be a variety over an algebraically closed field K. In defining $Z(X_0,t) = \prod \frac{1}{1-t^{\deg(y)}}$, we slip in a $t = q^{-x}$ when defining the global $\zeta$-function, to get $\zeta(X_0,s)$.
The local functions $Z$ are over $\Bbb F_p$ and can be expressed in terms of exp of trace of log of the Frobenius in the power series ring $\Bbb Q[[t]]$.
Why does it make the most sense to precompose each local factor by this $q^x =|\Bbb Z/q\Bbb Z|^x$ in the conglomerate product (obtaining the Hasse Weil zeta function)?