What is the point of Hecke $L$-series?

Question

Let $K$ be a number field and $\mathcal{O}_K$ its ring of integers. For an integral ideal $\mathfrak{m} \subset \mathcal{O}_K$ we let $J^{\mathfrak{m}}$ be the group of fractional ideals coprime to $\mathfrak{m}$.

A Hecke character $(\textrm{mod}\ \mathfrak{m})$ is a group character $\chi: J^{\mathfrak{m}} \to S^1$, satisfying some conditions.

A generalised Dirichlet character $\chi: J^{\mathfrak{m}}/P^{\mathfrak{m}} \to S^1$ is any group character of the ray class group $J^{\mathfrak{m}}/P^{\mathfrak{m}}$; $P^{\mathfrak{m}}$ being the subgroup of principal ideals congruent to $1\ (\textrm{mod}\ \mathfrak{m})$.

In his Class Field Theory, J. Neukirch introduces only generalised Dirichlet $L$-series, defined in terms of generalised Dirichlet characters. He uses these to prove:

1) The generalised Dirichlet density theorem.

2) That Artin $L$-series of Abelian extensions are in fact generalised Dirichlet $L$-series.

Then in his Algebraic Number Theory, he chooses instead to work with Hecke $L$-series, as opposed to generalised Dirichlet $L$-series. But he does not seem to derive any new results using this more general definition.

So my question is: What is the point of considering the more general Hecke $L$-series, as opposed to just restricting one's attention to generalised Dirichlet $L$-series?

Answer 1

The point is that a generalized Dirichlet character must have finite order (its domain is a finite group) but a Hecke character can have infinite order! See here for examples. In fact, the generalized Dirichlet characters are precisely the Hecke characters of finite order (that is a theorem, not a tautology).

Over $\mathbf Q$ it's hard to appreciate the need for general Hecke characters, since each Hecke character is closely related to some Dirichlet character. Roughly speaking, every Hecke character on $\mathbf Q$ looks like $\chi(n) = \chi'(n)|n|^{it_0}$ where $\chi'$ is a (primitive) Dirichlet character mod $m$ for some $m$ and $t_0 \in \mathbf R$. That means the $L$-function of $\chi$ is the $L$-function of $\chi'$ up to a vertical shift: $L(s,\chi) = L(s-it_0,\chi')$. So Hecke $L$-functions on $\mathbf Q$ are essentially the same thing as Dirichlet $L$-functions.

For a general number field $K$, Hecke characters $\chi$ on $K$ that are analogous to the ones just described on $\mathbf Q$ would be those where $\chi(\mathfrak a) = \chi'(\mathfrak a){\rm N}(\mathfrak a)^{it_0}$ for a Hecke character $\chi'$ on $K$ of finite order and $t_0 \in \mathbf R$. When $K \not= \mathbf Q$ there are many Hecke characters on $K$ not of that form. What does this mean in terms of Hecke $L$-functions? The $L$-function of a generalized Dirichlet character on $K$ is a factor of the zeta-function of some finite abelian extension of $K$, just as $L$-functions of (primitive) Dirichlet characters are factors of zeta-functions of cyclotomic fields. While the same Hecke $L$-function can arise from different Hecke characters (see here), when $K \not= \mathbf Q$ many Hecke $L$-functions are not related in any way to zeta-functions of finite abelian extensions (or really any finite extension) of $K$. Such $L$-functions are genuinely new things. The discussion here would be helpful.

The analytic continuation and functional equation of $L$-functions of all Hecke characters (originally proved by Hecke, then reproved by Tate in his thesis in 1950) is a much broader result than analytic continuation and functional equation of pieces of zeta-functions of number fields.