Valuation of exponential sums and the generic first slope for Artin–Schreier curves

We define the p-density of a finite subset D⊂Nr, and show that it gives a sharp lower bound for the p-adic valuations of the reciprocal roots and poles of zeta functions and L-functions associated to exponential sums over finite fields of characteristic p. When r=1, the p-density of the set D is the first slope of the generic Newton polygon of the family of Artin–Schreier curves associated to polynomials with their exponents in D.