Thank you both for your replies. Thanks for Brooker's paper, I was unaware of this result. @WillSawin I'm sorry for my lack of knowledge but which Hecke's theorem are you referring to? I guess another example would be an elliptic curve over $\mathbb{Q}$, since then both $M$ and its twist are modular, but it is perhaps wrong to say that the analytic properties of one is deduced from the other...

Alright, thank you for your help, I will try this.

Thank you very much for your clear answer. I do have some interest in the case where the character is quadratic. How would I go about computing the epsilon factor as you did? Do you have a reference that could be useful? Thank you.