|Revista Colombiana de Matemáticas|
Abstract. Let X be a complete, geometrically irreducible, algebraic curve defined over a finite field Fq and let (X ,t) be its zeta function [Ser1]. If X is a singular curve, two other zeta functions exist. The first is the Dirichlet series Z(Ca(X), t) associated to the effective Cartier divisors on X ; the second is the Dirichlet series Z(Div(X), t) associated to the effective divisors on X. In this paper we generalize F. K. Schmidt,s results on the rationality and functional equation of the zeta function (X ,t) of a non-singular curve to the functions Z(Ca(X), t) and Z(Div(X), t) by means of the singular Riemann-Roch theorem.Palabras claves. Zeta functions, finite fields, singular curves, generalized Jacobians, compactified Jacobians.