Zeta function of variety over positive characteristic function field vs. local zeta factor of variety over $\mathbb{F}_p$

Question

Let $X = Y \times_{\mathbb{F}_q} C$, with $Y, C / \mathbb{F}_q$ smooth projective varieties, $C$ a curve. Let $d = \dim_{\mathbb{F}_q} X$. We can consider the local zeta function $Z(X, t) = \prod\limits_{i = 0}^{2d} \det(1 - t \operatorname{Frob}_q | H^i(X_{\overline{F}_q}, \mathbb{Q}_\ell))^{(-1)^{i +1}}$, for $\ell, q$ coprime. Alternatively, we can base change $Y$ to the function field $k(C)$, and consider the Hasse–Weil zeta function of $Y$, which is the product over all places of $C$ of the local zeta function of the base change of $Y$ to the place. Is the Hasse–Weil zeta function of $Y$ the same as the local zeta function of $X$? What is their relationship?

Answer 1

The two zeta functions are the same. This is an immediate corollary of Milne, Etale Cohomology, proposition 13.8(c).

Reference:

Milne, J. S. Etale Cohomology (PMS-33). Princeton University Press, 1980. JSTOR, http://www.jstor.org/stable/j.ctt1bpmbk1.