  Given a Galois cover of curves over $\mathbb{F}_p$, we relate the $p$-adic
valuation of epsilon constants appearing in functional equations of Artin
L-functions to an equivariant Euler characteristic. Our main theorem
generalises a result of Chinburg from the tamely to the weakly ramified case.
We furthermore apply Chinburg's result to obtain a `weak' relation in the
general case. In the Appendix, we study, in this arbitrarily wildly ramified
case, the integrality of $p$-adic valuations of epsilon constants.
