Let $\chi : (\mathbb Z/f\mathbb Z)^\times \to K = \mathbb Q(\mu\_{\phi(f)})$ be a primitive Dirichlet character. Assume moreover that it is *not* quadratic, that is, $\chi^2$ is not the trivial character. Let $\pi\_1,\dots,\pi\_g$ be the primes lying over $2$ and $v\_1,\dots,v\_g$ be the corresponding valuations. Recall that:
$$L(0,\chi) = \frac1f\sum\_{n=1}^fn\chi(n)$$


Experimentally (upto conductor 200), I find that there always exists some $k$ such that $v\_k(L(0,\chi)) > 0$. Does anyone know a proof?


Note that it is not true that $2 | L(0,\chi)$. For instance, for for the character of conductor $5$ mapping $2 \to i$, we have $L(0,\chi) = (i+3)/5$. There are lots of other examples.


Note also that we do require the condition that $\chi$ is non quadratic. For instance, if $f = p \equiv 3 \pmod4$ and $\chi$ is quadratic, then:
$$pL(0,\chi) \equiv (p-1)/2 \equiv 1 \pmod 2.$$


I asked this question a few hours before on stackexchange but at the suggestion of someone, I am posting it here. I have deleted the question on stackexchange.



2-adic valuation of $L(0,\chi)$ for a Dirichlet character
Your question is entirely answered and with a lot of additional detail by Corollary 11.4.2 of my book Springer Graduate Texts in Math GTM 240. This must be in the literature outside of my book, but I do not know a reference.


