Let $\chi$ be a Hecke character on a number field $k$, where could I find a precise reference for the function equation of the $GL(1)$ L-functions $$L(s, \chi)?$$
I only find references for the case of $k = \mathbb{Q}$.
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1$\begingroup$ It depends on how explicit you want things, but a "standard" reference is usually to Table 5.3 of Deligne's paper "Valeurs de fonctions L", and for algebraic Hecke characters more particularly see Section 6 of Chapter Zero of Schappacher's book "Periods of Hecke Characters" (Springer LNM 1301) $\endgroup$– literature-searcherCommented Dec 5, 2018 at 9:59
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$\begingroup$ I'd say it depends on your definition of Hecke characters. You can look at the change of variable $x = y^{-1}$ in the adelic integral $\int_{k^\times \setminus \mathbb{A}_k^\times} \chi(x) |x|^s dx$ ($dx$ Haar measure on $\mathbb{A}_k^\times$) or look at $\sum_{a \in \mathbb{Z}^n} \psi(b^\top a) P(a) \exp(-\pi \sum_j y_j |\sigma_j(b^\top a)|^2), O_k = b^\top \mathbb{Z}^n,\psi$ a character of $O_K/I^\times$ (if primitive so is its discrete Fourier transform), $ P$ an homogeneous polynomial, and apply the Poisson summation formula to make a Hecke series over the different ideal appear. $\endgroup$– reunsCommented Dec 5, 2018 at 11:20
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$\begingroup$ Hecke's original papers are not bad! $\endgroup$– paul garrettCommented Dec 5, 2018 at 17:03
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$\begingroup$ @paulgarrett Where can I find Hecke's original papers? This is not the first time they are pointed out as a relevant reference. I am mainly interested on quadratic Hecke characters, say $\chi_m$ for an ideal $m$, and I would like to have the explicit dependence in $m$ of the extra factors appearing in the functional equation. $\endgroup$– TheStudentCommented Dec 6, 2018 at 1:38
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$\begingroup$ The relevant Hecke papers are: [Hecke~1918] E. Hecke, {\it Eine neue Art von Zetafunktionen und ihre Beziehungen der Verteilung der Primzahlen}, Math. Z. {\bf 1} no. 4 (1918), 357-376. [Hecke~1920] E. Hecke, {\it Eine neue Art von Zetafunktionen und ihre Beziehungen der Verteilung der Primzahlen}, Math. Z. {\bf 6} no. 1-2 (1920), 11-51. [Hecke~1922/24] E. Hecke, {\it Analytische Funktionen und algebraische Zahlen, I, II}, Abh. Math. Sem. Hamburg {\bf 1} no. 1 (1922), 102-126; {\bf 3} no. 1-2 (1924), 213-236. $\endgroup$– paul garrettCommented Dec 6, 2018 at 1:43
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