Definition 63.19.1. 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 $\mathcal{F}$*.

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