TITLE: von Staudt-Clausen for other special values
QUESTION [6 upvotes]: 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.

REPLY [3 votes]: For an elliptic curve over $\mathbb{Q}$, the analogous question is what is the denominator of the algebraic part of $L(E,\chi,1)$ as $\chi$ varies in the Dirichlet characters. 
First if $\chi$ is trivial, then the precise question is the denominator of $L(E,1)/\Omega^+$ where $\Omega^+$ is the least positive period of the Néron differential. If it is non-zero, then the only odd primes $p$ that can divide this denominator are those for which the Galois module $E[p]$ is reducible. But even those need not arise depending on which elliptic curve in the isogeny class is taken. BSD would tell you what the denominator is; some divisor of the torsion order.
Now for other twists the same is true, only those odd prime can divide the denominator (and very often the values are integers in fact). Since these values are integral sums of modular symbols it all boils down to study the denominator of modular symbols.
For more general modular forms one has to be careful when making the question precise. It depends a lot on what period is chosen and there may not always be a canonical best choice.