Let $K=\Bbb Q(e^{2\pi i/m})$ be a cyclotomic field. I have frequently seen the assertion that its Dedekind zeta function $\zeta_K(s)$ factors into Dirichlet $L$-functions: $$ \zeta_K(s) = \prod_{\chi\pmod m} L(s,\chi). $$ However, in Neukirch's Algebraic Number Theory, Proposition 5.12 states instead that $$ \zeta_K(s) = \prod_{\mathfrak p\mid m} (1-\mathfrak N(\mathfrak p)^{-s})^{-1} \prod_{\chi\pmod m} L(s,\chi). $$ I'm confused by this apparent discrepancy and could use some remedial help. So:
- Does $L(s,\chi)$ refer to a possibly imprimitive character $\chi\pmod m$, or to a primitive character modulo a divisor of $m$?
- Are the factors for $\mathfrak p\mid m$ present in the formula or not?
- If they are present, can one evaluate them in terms of $m$ alone and not the arithmetic of $K$?
- Is there one precise and clear statement that would serve as a suitable reference for me to cite?