Local Zeta Function

In mathematics, the local zeta function Z(V, s) (sometimes called the congruent zeta function or the Hasse-Weil zeta function) is defined as Z ( V, s) = exp ⁡ ( ∑ k = 1 ∞ N k k ( q − s) k) {\displaystyle Z(V,s)=\exp \left(\sum _{k=1}^{\infty }{\frac {N_{k}}{k}}(q^{-s})^{k}\right)} where V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements and Nk is the number of points of V defined over the finite field extension Fqk of Fq.

The first study of these functions was in the 1923 dissertation of Emil Artin.