On the invariants of L-functions of degree 2, I: Twisted degree and internal shift

This is the first part of a series of papers where the behavior of the invariants under twist by Dirichlet characters is studied for L-functions of degree 2. Here we show, under suitable conditions, that degree and internal shift remain unchanged under twist. The ultimate goal of the series is to prove a general version of Weil converse theorem with minimal assumptions on the shape of the functional equation of the twists.