Motivation for Hecke characters The context is the definition of Hecke Größencharakter:
http://en.wikipedia.org/wiki/Hecke_character
This is supposed to generalize the Dirichlet $L$-series for number fields. Dirichlet characters are characters of the multiplicative groups of $\mathbb Z/p\mathbb Z$. An appropriate generalization would be instead to consider characters of the multiplicative group of $\mathcal O_K/\cal P$ where $\mathcal P$ is a prime ideal in the ring of integers of a number field $K$.
But Hecke Größencharakter goes to more trouble than this. It brings in ideles and such for a more complicated generalization. Why is this necessary?

One reason for the greater generality is that Hecke
characters do more than describe Abelian extensions of number fields
(essentially Dirichlet characters describe Abelian extenions
of $\mathbb Q$). For instance the L-function of an elliptic
curve with complex multiplication is the L-function of a Hecke
character of infinite order.
