This is surely well known and I might even have seen it before:
The Riemann zeta function can be expressed as a Mellin transform:
$$\Gamma(s)\zeta(s) = \int_0^\infty x^{s-1}\frac{1}{e^x - 1}dx.$$
How does one express the Dedekind zeta function as a Mellin transform, particularly for totally real number fields or CM fields? I believe in these two cases, it comes from modular forms - which ones?
What happens (or is expected to happen) in the more general case of L-functions associated to the various characters on these fields?