Let $M$ be a pure motive over a number field and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic continuation and a functional equation relating $\Lambda(M, s)$ and $\Lambda(M^\vee, 1-s)$ as conjectured. Let $\chi$ denote a quadratic Dirichlet character.

Can one deduce the analytic continuation and functional equation of the twisted $L$-function $\Lambda(M, \chi, s)$ from the properties of $L(M, s)$?

Thank you in advance for your help.

Let $M$ be a pure motive over $\mathbb{Q}$ and consider the (completed) $L$-function $\Lambda(M, s)$ attached to its $\ell$-adic realization. Let us assume that this $L$-function admits analytic continuation and a functional equation relating $\Lambda(M, s)$ and $\Lambda(M^\vee, 1-s)$ as conjectured. Let $\chi$ denote a quadratic Dirichlet character.

Can one deduce the analytic continuation and functional equation of the twisted $L$-function $\Lambda(M, \chi, s)$ from the properties of $L(M, s)$?

Thank you in advance for your help.