In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as

where V is a non-singular n-dimensional projective algebraic variety over the field F_q with q elements and N_k is the number of points of V defined over the finite field extension F_q^k of F_q.

Making the variable transformation t = q^−s, gives

as the formal power series in the variable ${\displaystyle t}$.

Equivalently, the local zeta function is sometimes defined as follows:

In other words, the local zeta function Z(V, t) with coefficients in the finite field F_q is defined as a function whose logarithmic derivative generates the number N_k of solutions of the equation defining V in the degree k extension F_q^k.

Given a finite field F, there is, up to isomorphism, only one field F_k with

for k = 1, 2, ... . When F is the unique field with q elements, F_k is the unique field with ${\displaystyle q^{k}}$ elements. Given a set of polynomial equations — or an algebraic variety V — defined over F, we can count the number