Hi!


This is my first question here.


In studying automorphic form, I am wondering the relation of critical L-values of some representation and its twisted representation by a character.


For example, let $F$ be a global number field and $G$ be an algebraic group over $F$. And $\pi$ is an irreducible cuspidal automorphic representation of $G(A\_F)$ and $\chi$ is a unitary character of $G(A\_F)$.


Then I am very curious whether there is some relationship between $L(\frac{1}{2},\pi \otimes \chi)$ and $L(\frac{1}{2},\pi)$. For example, their nonvanishing is equivalent especially when $\pi$ is character?


Since I am very beginner just started in this area and it might be a stupid question to many experts here. But, any help or remark will be greatly appreciated!



A question on twisted  L-function
The answer is no. Consider G=Gl(n). We have $L(s,\pi)=L(s-s',\pi\otimes|det|^{s'})$.
The absolute value is the adelic norm. In general,eg in Tate's thesis, the parameter s is sometimes avoided for this reason.


So your conjecture violates GRH. For n=1 and $\pi$ trivial over the rational numbers gives you a concrete counter example, because there exist zeros on the critical line.


