I recently stumbled upon the slides of a talk given by Kedlaya, in which the following appears:

For $X$ of finite type over $F_q$, a Weil cohomology theory, mapping $X$ to certain vector spaces $H^i(X)$ over a field of characteristic zero, provides a spectral interpretation of $\zeta(X,s)$ via the formula: $$\zeta(X,s)=\prod_i \det(1-q^{-s}\text{Frob}_q,H^i(X))^{(-1)^{i+1}}$$

Where $\zeta(X,s)$ is the arithmetic zeta function of a scheme $X$ of finite type over $\mathbb{Z}$.

This is *incredibly* intriguing to me, and seems to be intimately related to some objects I'm trying to study - however, this slide really doesn't give me enough information to pull out some *juicy math*. Where does this formula come from (it seems very close to but not equivalent to the product in the definition of an Artin L-Function), and how does it provide a spectral interpretation of the arithmetic zeta function?

If the answer is too big to be self-contained, I'd also appreciate a reference or some buzzwords to google.