Hasse–Weil conjecture for motivic pieces of curves

Let $C$ be a curve over a number field $K$ with an action by a finite group $G$, let $\tau$ be an irreducible representation of $G$, and let $C^\tau$ be the corresponding motivic piece. Write $L(C^\tau,s)$ for its $L$-function, $\tau^*$ for the dual representation, and $\tau$'s completed $L$-function as $Lambda(C^\tau,s)$, with root number $w(C^\tau)$. Hasse–Weil conjecture. $L(C^\tau,s)$ admits meromorphic continuation to $\mathbb{C}$, and

$$\Lambda(C^\tau,s)=w(C^\tau)\cdot\Lambda(C^{\tau^*},2-s).$$

This is the expected analytic continuation and functional equation for the $L$-functions attached to motivic pieces; the paper treats it as part of the conjectural framework and does not establish it in general.