Let K be a function field over a finite field and let L be a finite extension of K. It is well-known by the Riemann hypothesis of curves that the zeros of the zeta function of L all lie on a circle. It is interesting to study how the zeros are distributed on the circle when L is taken over a family of function fields, and various families have been considered. In this work we take all abelian extensions of K with a given Galois group as our family, ordered by the conductor. We prove that as the absolute norm of the conductor goes to infinity, for any fixed arc on the circle, the number of zeros lying in the arc is proportional to the length of the arc, and after a suitable normalization it converges to a standard Gaussian distribution.