Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$: $$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$ where $h,R,w$ stand for the size of the class group, the regulator and the size of the subgroup of roots of unity.

On the other hand, we have the decomposition: $$\zeta_K(s) = \prod_\chi L(s,\chi)$$ where the product is over suitable Dirichlet characters and moreover, $L(0,\chi) = -B_{1,\chi}$ - the generalized Bernoulli numbers.

On the other hand, $R(K)$ should not be algebraic and there should be a corresponding transcendental contribution from $L(0,\chi)$ and the explicit formula for $B_{1\chi}$ shows that this comes only from those $\chi$ such that $L(s,\chi) = 0. $

**Question 1:** In the case that $L(s,\chi) = 0$, is it possible to say what the first non zero Taylor coefficient is?

**Question 2:** Can we decompose (even if it is only conjecturally) $h(K),R(K),w(K)$ into factors corresponding to each Dirichlet character appearing in the decomposition of $\zeta_K$ . What about the order of vanishing of $L(0,\chi)$?

Cyclotomic Fieldsor Coates and Sujatha'sCyclotomic Fields and Zeta Values. $\endgroup$ – David Loeffler Feb 15 '18 at 15:09