In number theory, the local zeta function Z(V, s) (sometimes called the congruent zeta function) is defined as where Nm is the number of points of V defined over the mth cyclotomic field extension Fqm of Fq, and V is a non-singular n-dimensional projective algebraic variety over the field Fq with q elements. By the variable transformation u = q−s, then it is defined by as the formal power series of the variable [math]\displaystyle{ u }[/math]. Equivalently, the local zeta function sometimes is defined as follows: In other word, the local zeta function Z(V, u) with coefficients in the finite field Fq is defined as a function whose logarithmic derivative generates the numbers Nm of the solutions of equation, defining V, in the m degree extension Fqm.