How do I understand Hecke characters? For example, is there a bijection between Hecke characters and something? For example, if a Hecke character factors through a ray class group, by Artin reciprocity, we would have a 1-1 correspondence with characters of finite abelian Galois groups with restricted ramification.
I'm aware of Understanding Hecke Characters as Extension of Dirichlet Characters, but this does not answer my question. Motivation for Hecke characters is a bit more helpful. For example, what is the interpretation of infinity types?