Let $\mathbf{G}$ be a reductive group which enjoys all the nice properties a reducive group can dream of. Fix $(\mathbf{G},X)$ a Shimura datum associated with it and assume that if $K\leq\mathbf{G} $ is neat then the locally symmetric space associated has a model $S_{K}$ over the reflex field.
We then conjecture that in the case of a cuspidal cohomological representation of $\mathbf{G}$ of level $K$, the Galois representation given by the Langlands correspondence is (perhaps up to a certain multiplicity) the Galois-module part of the $\pi$-isotypic component of the cohomology of $S_{K}$, and it is a theorem in some examples of group $\mathbf{G}$.
But the L function of a cuspidal representation depends on the choice an embeding $\mathbf{G}^{L}\overset{r}{\rightarrow} \mathbf{Gl}_{n}$, and the former Galois representation do not depend on such choice.
This observation suggests that there should be a canonical choice of $r$, right?
I convinced myself before that the convention was to choose the L-function wich appear in the Hasse-Weil zeta function of $S_{K}$, but recently I understood that the Hasse-Weil zeta function of $S_{K}$ is difficult to compute.
Question How to choose $r$? Is it related to the Hasse-Weil zeta function of $S_{K}$?
Question Any references?