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  "body": "\n## Summary\n\nThe Hasse–Weil zeta function of a variety over a number field is an Euler product.\nIt packs together the numbers of points of the variety modulo all primes.\nFor a point it is the [[Riemann zeta function|1XWBY4N0VT]] or the [[Dedekind zeta function|KXS5DQDFYD]].\nFor an [[elliptic curve|427N6S6QQW]] it is built from the $L$-function of the [[Birch and Swinnerton-Dyer conjecture|ZWEFBG40H9]].\nIts analytic continuation, functional equation and special values are the subject of deep conjectures, some of them proved through the [[modularity theorem|CS1NVJBZMH]].\n\n## Explanations\n\n### L1 human\n\nRiemann's zeta function can be written as a product over all primes $p$ of the factors $(1 - p^{-s})^{-1}$.\nThe Hasse–Weil zeta function does the same for a system of polynomial equations with integer coefficients.\nFor each prime $p$, one counts the solutions modulo $p$ and in the finite fields of order $p^2, p^3, \\ldots$, and turns these counts into a local factor.\nThe product of the local factors over all primes is the Hasse–Weil zeta function.\nFor an elliptic curve $E$ over $\\Q$, it equals $\\zeta(s)\\zeta(s-1)/L(E, s)$, up to finitely many factors at the bad primes.\n",
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