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  "body": "\n## Definition\n\nLet $X$ be a scheme of finite type over $\\Z$, of Krull dimension $d$.\nLet $\\abs{X}$ be its set of closed points; the residue field $\\kappa(x)$ of each $x \\in \\abs{X}$ is a [[finite field|0ZXZ82FTX2]], and let $N(x) = \\#\\kappa(x)$.\nThe *Hasse–Weil zeta function* of $X$ is\n$$\\zeta_X(s) = \\prod_{x \\in \\abs{X}} \\frac{1}{1 - N(x)^{-s}},$$\nan Euler product that converges absolutely for $\\mathrm{Re}(s) \u003e d$.\n\nNow let $V$ be a smooth projective variety over $\\Q$.\nChoose a finite set of primes $S$ and a model $X$ of $V$ that is smooth and projective over $\\Z[1/S]$.\nFor $p \\notin S$, let $X_p$ be the fibre of $X$ over $\\F_p$, and let\n$$Z(X_p, t) = \\exp\\left( \\sum_{m \\ge 1} \\frac{\\#X_p(\\F_{p^m})}{m} \\, t^m \\right).$$\nThe zeta function of $V$ is the product of the local factors $Z(X_p, p^{-s})$ over $p \\notin S$.\nChanging the model changes only finitely many factors.\n\n## Remarks\n\nThe two forms agree: the local factor $Z(X_p, p^{-s})$ is the product of $(1 - N(x)^{-s})^{-1}$ over the closed points $x$ of $X$ lying over $p$.\n\nExample: for $X = \\mathrm{Spec}\\, \\Z$, the closed points are the primes $p$ with $N(p) = p$, so $\\zeta_X(s) = \\prod_p (1 - p^{-s})^{-1} = \\zeta(s)$, the [[Riemann zeta function|1XWBY4N0VT]], which converges for $\\mathrm{Re}(s) \u003e 1 = d$.\nFor $X = \\mathbb{A}^1_{\\Z}$, one finds $\\zeta_X(s) = \\zeta(s - 1)$, since $\\#\\mathbb{A}^1(\\F_{p^m}) = p^m$.\n\nNon-example: the [[Dirichlet series|8MWTEF6462]] $\\sum_{n \\ge 1} (-1)^{n} n^{-s}$ is not of this form, since its coefficients take negative values, while expanding an Euler product of factors $(1 - N(x)^{-s})^{-1}$ gives nonnegative integer coefficients.\n\nFor an elliptic curve $E$ over $\\Q$ with a good model away from its conductor, the local factor at a good prime $p$ is $\\dfrac{1 - a_p p^{-s} + p^{1 - 2s}}{(1 - p^{-s})(1 - p^{1-s})}$.\nThe Hasse–Weil conjecture predicts meromorphic continuation to $\\C$ and a functional equation; it is known for elliptic curves over $\\Q$ by the modularity theorem.\n",
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