Motivic pieces of curves: $L$-functions and periods

Given a curve $C$ over a number field $K$ equipped with the action of a finite group $G$ by $K$-automorphisms, one obtains a factorisation of $L(C,s)$ into a product of $L$-functions of `motivic pieces of curves' associated to irreducible $G$-representations. We describe an algorithm for explicitly computing values of these $L$-functions, demonstrating implementations in the cases of certain curves with actions by $C_3$, $C_4$ and $D_{10}$. We explain how this algorithm can be used to factor $L$-functions of curves with endomorphisms of Hecke type. Towards applications, we explicitly formulate and numerically verify a version of Deligne's Period Conjecture for hitherto-uninvestigated $L$-functions arising from motivic pieces of superelliptic curves.