On slopes of L-functions of Z_p-covers over the projective line

Let P:⋯→C₂→C₁→P¹ be a Z_p-cover of the projective line over a finite field of cardinality q and characteristic p which ramifies at exactly one rational point. We study the q-adic valuations of the reciprocal roots in C_p of L-functions associated to characters of the Galois group of P. We show that for all covers P such that the genus of C_n is a quadratic polynomial in pⁿ for n large, the valuations of these reciprocal roots are uniformly distributed in the interval [0,1]. Furthermore, we show that for a large class of such covers P, the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.