= Artin L-function order at one
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{title2=$\operatorname{ord}_{s=1}L_K(\chi,s)=-\langle\chi,1_G\rangle_G$}
Assume that every nontrivial one-dimensional <Artin L-function> over the intermediate fields is holomorphic and nonzero at one, while the trivial one is the base <Dedekind zeta function> with a simple pole. By <Brauer induction>, $\chi=\sum_jn_j\operatorname{Ind}_{H_j}^G\psi_j$ with integers $n_j$ and <linear characters> $\psi_j$. Compatibility of <Artin L-functions> with <induced representations> expresses $L_K(\chi,s)$ as the corresponding product of $L_{L^{H_j}}(\psi_j,s)^{n_j}$. Its order at one is $-\sum_jn_j\langle\psi_j,1_{H_j}\rangle$. <Frobenius reciprocity> turns this sum into $-\langle\chi,1_G\rangle$. Hence every nontrivial <irreducible character> gives order zero and a holomorphic nonzero function near one. This local conclusion does not assert global Artin holomorphy.
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