Let $G$ be a split connective reductive group over $k=\mathbb F_q$, then the affine Grassmannian $X=Gr_G$ is representable by an ind-projective strict ind-scheme over $k$. (That is, there exists an inductive system of projective schemes over $\mathbb F_q$, and the maps are closed inclusions.)

Is it possible to define zeta function for $Gr_G$? And what about the analog of Weil conjecture for $Gr_G$ with relations to the cohomology of $Gr_G$?

The naive example: $Gr_{GL_n}=\text{colim}_{i} \ Gr^{(i)}_{GL_n}$, one computes $\# Gr^{(i)}_{GL_n}(\Bbb F_q)= \sum_{d=0}^{\infty} a_{d,i} q^d$ is a polynomial of $q$. The limit of $\frac{a_{d,i}}{2i+1} (i \rightarrow +\infty)$ exists, and we call the limit $a_d$ and form the formal series $\sum_{d=0}^{\infty}a_dq^d \in \Bbb Z[[q]]$ . Then one define the zeta function as an element in $\Bbb Q[[q,t]]$ by the usual definition using exponential.