© 2015, Springer-Verlag Berlin Heidelberg. We fix a monic polynomial (Formula presented.) over a finite field and consider the Artin-Schreier-Witt tower defined by f(x); this is a tower of curves (Formula presented.) , with total Galois group (Formula presented.). We study the Newton slopes of zeta functions of this tower of curves. This reduces to the study of the Newton slopes of L-functions associated to characters of the Galois group of this tower. We prove that, when the conductor of the character is large enough, the Newton slopes of the L-function form arithmetic progressions which are independent of the conductor of the character. As a corollary, we obtain a result on the behavior of the slopes of the eigencurve associated to the Artin-Schreier-Witt tower, analogous to the result of Buzzard and Kilford.