Hasse–Weil conjecture for smooth projective varieties

Let $X$ be a smooth, projective variety over a number field $F$ with good reduction outside a finite set of primes $S$. Its global Hasse–Weil zeta function is

$$\zeta_X(s)=\prod_x\frac{1}{1-N(x)^{-s}},$$

where the product runs over the closed points of a smooth proper integral model of $X$ over $\mathcal{O}_F[1/S]$.

Hasse–Weil conjecture. The function $\zeta_X(s)$ extends to a meromorphic function of $\mathbf C$. There exists a positive real number $A\in\mathbf R^{>0}$, non-zero rational functions $P_v(T)$ for $v\mid S$, and infinite Gamma factors $\Gamma_v(s)$ for $v\mid\infty$ such that

$$\xi(s)=\zeta_X(s)\cdot A^{s/2}\cdot\prod_{v\mid\infty}\Gamma_v(s)\cdot\prod_{v\mid S}P_v(N(v)^{-s})$$

satisfies the functional equation

$$\xi(s)=w\cdot\xi(\dim X+1-s),\qquad w=\pm1.$$

This is the Hasse–Weil conjecture for the zeta function of a smooth projective variety. The source presents it in an account of results proving the conjecture for abelian surfaces and genus $2$ curves over totally real fields; its general status is not resolved by the supplied text.