The Stacks project

Definition 64.19.1. Let $X$ be a scheme of finite type over a finite field $k$. Let $\Lambda$ be a finite ring of order prime to the characteristic of $k$ and $\mathcal{F}$ a constructible flat $\Lambda$-module on $X_{\acute{e}tale}$. Then we set

where $|X|$ is the set of closed points of $X$, $\deg x = [\kappa (x): k]$ and $\bar x$ is a geometric point lying over $x$. This definition clearly generalizes to the case where $\mathcal{F}$ is replaced by a $K \in D_{ctf}(X, \Lambda )$. We call this the $L$-function of $\mathcal{F}$.

Definition 64.19.3. Now assume that $\mathcal{F}$ is a $\mathbf{Q}_\ell$-sheaf on $X$. In this case we define

Note that this product converges since there are finitely many points of a given degree. We call this the $L$-function of $\mathcal{F}$.