Hasse–Weil zeta function ($Z(V,s)$)

The Hasse–Weil zeta function of an algebraic variety over a number field is assembled from local point-counting factors at finite places. At a good place with residue field of size $q$, the factor is $Z(V_q,q^{-s})$, where $Z(V_q,T)=\exp ({\sum }_{n\ge 1}\#V_q({\mathbb{F}}_{q^n})T^n/n)$. Defining all bad-place factors requires a specified integral model or convention. The zeta function of an elliptic curve over a finite field is a local factor, rather than this global product.