On slopes of $L$-functions of $\mathbb{Z}_p$-covers over the projective line

Let $\mathcal{P}: \cdots \rightarrow C_2\rightarrow C_1\rightarrow {\mathbb P}^1$ be a $\mathbb{Z}_p$-cover of the projective line over a finite field of cardinality $q$ and characteristic $p$ which ramifies at exactly one rational point, and is unramified at other points. In this paper, we study the $q$-adic valuations of the reciprocal roots in $\mathbb{C}_p$ of $L$-functions associated to characters of the Galois group of $\mathcal{P}$. We show that for all covers $\mathcal{P}$ such that the genus of $C_n$ is a quadratic polynomial in $p^n$ 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 $\mathcal{P}$, the valuations of the reciprocal roots in fact form a finite union of arithmetic progressions.