For the arithmetic zeta function of (say) a nonsingular projective variety $X$, one has the following Euler product
\begin{equation} \zeta_X(s) = \prod_{p\ \mbox{prime}}\zeta_{X\vert\mathbb{F}_p}(s), \end{equation} where $\zeta_{X \vert \mathbb{F}_p}(s)$ is the local zeta function given by $\exp(\sum_{m = 1}^\infty \frac{\vert X_{p^m} \vert}{m}p^{-sm})$ (here, $X_{p^m}$ is the set of $p^m$-rational points of $X$). For any field $\mathbb{F}_q$, one also has a local zeta function for $X$, being $\zeta_{X \vert \mathbb{F}_q}(s) = \exp(\sum_{m = 1}^\infty \frac{\vert X_{q^m} \vert}{m}q^{-sm})$.

My question is: suppose one now considers the product \begin{equation} \prod_{p \ \mbox{prime}}\zeta_{X\vert\mathbb{F}_{p^r}}(s), \end{equation} for a fixed positive integer $r$, is there also a nice identity involving (arithmetic) zeta functions?