We have the relation $\zeta_{X\times Y}(s) = \zeta_X(s)* \zeta_Y(s)$ where $*$ is the Witt product in the Witt ring of $\mathbb Z[[t]]$. For any commutative ring A, the (big) Witt ring $W(A)$ is defined by:

- Under addition, $W(A), +$ is isomorphic to the group $(1 + tA[[t]],\times)$
- The multiplication $*$ is uniquely determined by $$(1-at)^{-1}(1-bt)^{-1} = (1-abt)^{-1}.$$

That this is indeed true can be verified on each Euler product: the $k$-points of $X\times Y$ are in bijection with pairs of $k$-points on $X,Y$. In fact, the same relation holds if we only assume that $Z$ is a fiber bundle over $X$ with fiber $Y$ by the same fact about points: to give a point $P$ on $Z$ is to give a point on $X$ (the image of $P$ under projection) and a point on the fiber.

A reference for this is https://arxiv.org/abs/1407.1813 and can also be found on the Wikipedia page: https://en.wikipedia.org/wiki/Arithmetic_zeta_function

[The reference only talks about this formula over finite fields but I think it holds for the zeta function of any scheme $X$ finite type over $\mathbb Z$ because such a zeta function is the product of a zeta function $X/\mathbb F_p$ over every finite field. As an example, the Riemann zeta function is the product of the zeta function of a point over every $\mathbb F_p$. Since the formula is proven Euler product by euler product, this is not an issue.