Let $X$ be a smooth projective variety of dimension $n$ over $\mathbb{F}_q$. Then: **(W1) Rationality.** $Z_X(t)$ is a rational function in $t$. **(W2) Functional equation.** We have \begin{align*} Z_X\!\left(\frac{1}{q^n t}\right) = \pm\, q^{nE/2} t^E \cdot Z_X(t), \end{align*} where $E = \chi(X_\mathbb{C})$ is the topological Euler characteristic of the complex variety. **(W3) Riemann hypothesis.** The rational function $Z_X(t)$ factors as \begin{align*} Z_X(t) = \frac{P_1(t) P_3(t) \cdots P_{2n-1}(t)}{P_0(t) P_2(t) \cdots P_{2n}(t)}, \end{align*} where $P_0(t) = 1 - t$, $\;P_{2n}(t) = 1 - q^n t$, and for $0 < i < 2n$, \begin{align*} P_i(t) = \prod_j (1 - \alpha_{ij} t) \end{align*} with $\alpha_{ij}$ algebraic integers satisfying $|\alpha_{ij}| = q^{i/2}$. Here $\alpha_{ij}$ are the eigenvalues of Frobenius on the $i$-th étale cohomology $H^i_\mathrm{et}(X, \mathbb{Q}_\ell)$; they have absolute value $q^{i/2}$, while the roots of $P_i(t)$ are $\alpha_{ij}^{-1}$ and have absolute value $q^{-i/2}$. **(W4) Betti numbers.** $\deg P_i = h^i(X_\mathbb{C};\, \mathbb{Q})$, the $i$-th Betti number of $X_\mathbb{C}$.