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Lef $K$ be an algebraic number field of degree $[K:\mathbb{Q}]=n$. For simplicity suppose $K$ is totally real. Define $f(s) = \zeta_K(s) \zeta(1-s)^{n-1}$ where $\zeta = \zeta_{\mathbb{Q}}$. From the Class number formula and the functional equation of the zeta function, the order of $s=0$ as zero of $\zeta_K$ is $n-1$ and $\zeta$ has simple pole at $s=1$ with residue $1$. So the limit $\lim_{s\to 0} f(s)$ exists and $f(s)$ is holomorphic function on neiberhood of $s=0$.

Is there a way to calculate the derivative $f'(0)$ ?

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Up to more elementary things, the quantity in question is the zero-th Laurent coefficient of $\zeta_K(s)$ at $s=1$, which is a kind of generalized Euler constant. If $K$ is abelian over $\mathbb Q$, this Euler constant is the classical one, multiplied by values $L(1,\chi)$ for Dirichlet characters such that the product of $\zeta(s)$ and these $L$-functions is the zeta function of $K$. – paul garrett Aug 23 '13 at 16:38
And the equivalent problem pointed out by paul garrett was discussed in this question:… – Lucia Aug 24 '13 at 2:22
@paulgarrett Thank you ! – colge Aug 24 '13 at 10:47

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