The von Staudt-Clausen theorem expresses that the Bernoulli numbers' denominators have a very special form (see the wikipedia page on the theorem for more details).

What interests me is that those Bernoulli numbers have a link with the Riemann $\zeta$ function (also discussed in this other question) : $$\forall n>1, \zeta(1-n)=\frac{-B_n}n$$ (and through the functional equation, likewise to the right -- with an explicit factor with $\pi$ powers)

I interpret this as an example of an arithmetical function with an arithmeticity result for some special values, where a theorem gives results on denominators.

Here comes the question now the big picture is in place : there are also arithmeticity results for special values of $L$ functions of Dirichlet characters. Are there results similar to the von Staudt-Clausen theorem in this case? And for other $L$-functions for which arithmeticity results do exist (say, elliptic modular forms)?

Notice that I checked the proof of the theorem for the Bernoulli numbers, and the one I found (in Hardy&Wright's "Introduction to the theory of numbers") uses the definition of the sequence as coefficients of a series expansion... something which isn't applicable as far as I know in the setting(s) of my question.