Let $C$ be a curve over a finite field and let $\rho$ be a nontrivial representation of $\pi_1(C)$. By the Weil conjectures, the Artin $L$-function associated to $\rho$ is a polynomial with algebraic coefficients. Furthermore, the roots of this polynomial are $\ell$-adic units for $\ell \neq p$ and have Archemedian absolute value $\sqrt{q}$. Much less is known about the $p$-adic properties of these roots, except in the case where the image of $\rho$ has order $p$. We prove a lower bound on the $p$-adic Newton polygon of the Artin $L$-function for any representation in terms of local monodromy decompositions. If time permits, we will discuss how this result suggests the existence of a category of wild Hodge modules on Riemann surfaces, whose cohomology is naturally endowed with an irregular Hodge filtration.