Hasse–Weil zeta function

Remarks

The 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 $p$.

Example: 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, which converges for $\mathrm{Re}(s) > 1 = d$. For $X = \mathbb{A}^1_{\Z}$, one finds $\zeta_X(s) = \zeta(s - 1)$, since $\#\mathbb{A}^1(\F_{p^m}) = p^m$.

Non-example: the Dirichlet series $\sum_{n \ge 1} (-1)^{n} n^{-s}$ 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.

For an elliptic curve $E$ over $\Q$ with a good model away from its conductor, the local factor at a good prime $p$ is $\dfrac{1 - a_p p^{-s} + p^{1 - 2s}}{(1 - p^{-s})(1 - p^{1-s})}$. The Hasse–Weil conjecture predicts meromorphic continuation to $\C$ and a functional equation; it is known for elliptic curves over $\Q$ by the modularity theorem.