Definition 63.19.3. Now assume that $\mathcal{F}$ is a $\mathbf{Q}_\ell$-sheaf on $X$. In this case we define

$L(X, \mathcal{F}) := \prod \nolimits _{x \in |X|} \det (1 - \pi _ x^*T^{\deg x} |_{\mathcal{F}_{\bar x}})^{-1} \in \mathbf{Q}_\ell [[T]].$

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

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