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.