Hasse–Weil zeta function

Summary

The Hasse–Weil zeta function of a variety over a number field is an Euler product. It packs together the numbers of points of the variety modulo all primes. For a point it is the Riemann zeta function or the Dedekind zeta function. For an elliptic curve it is built from the $L$-function of the Birch and Swinnerton-Dyer conjecture. Its analytic continuation, functional equation and special values are the subject of deep conjectures, some of them proved through the modularity theorem.