Editing Local zeta function (section)

==Motivations==

The relationship between the definitions of ''G'' and ''Z'' can be explained in a number of ways. (See for example the infinite product formula for ''Z'' below.) In practice it makes ''Z'' a [[rational function]] of ''t'', something that is interesting even in the case of ''V'' an [[elliptic curve]] over a finite field.

The local ''Z'' zeta functions are multiplied to get global ''<math>\zeta</math>'' zeta functions,

<math>\zeta = \prod Z</math>

These generally involve different finite fields (for example the whole family of fields '''Z'''/''p'''''Z''' as ''p'' runs over all [[prime number]]s).

In these fields,  the variable ''t'' is substituted by ''p<sup>−s</sup>'', where ''s'' is the complex variable traditionally used in [[Dirichlet series]]. (For details see [[Hasse–Weil zeta function]].)

The global products of ''Z'' in the two cases used as examples in the previous section therefore come out as <math>\zeta(s)</math> and <math>\zeta(s)\zeta(s-1)</math> after letting <math>q=p</math>.