Let $P: ... \to C_2 \to C_1 \to P^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of characteristic $p$ which ramifies at exactly one rational point. In this talk, we study the $p$-adic Newton slopes of L-functions associated to characters of the Galois group of $P$. It turns out that for covers $P$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ for $n$ large, the Newton slopes are uniformly distributed in the interval $[0,1]$. Furthermore, for a large class of such covers $P$, these slopes behave in an even more regular way. This is joint work with Hui June Zhu.