The L-functions associated with finite characters of number fields (often called Dirichlet charaters
or finite Hecke characters)
were considered by Hecke in a series of papers beginning in 1917.
In the first paper of this series, *Über die Zetafunktion*
*beliebiger algebraischer Zahlkörper* [Gött. Nachr. 1917 77-89 (1917)], he refers to $\zeta_k(s)$ as the "Dirichlet-Dedekindsche Zetafunktion".
Here he gives the functional equation for the Dedekind zeta-function of any number field.
In the second
paper *Über eine neue anwendung der Zetafunktionnen
auf die arithmetik der Zahlkörper* also in 1917, he refers to it as
"der Dedekindschen Funktion $\zeta_k(s)$." This paper ends with the remark that
for abelian extensions $\zeta_k(s)$ can be factored as a product of Dirichlet L-functions.
In the third paper in the series, he refers again to
"Dirichlet-Dedekindsche Funktion $\zeta_k(s)$".
Here he proves the functional equation for L-functions of finite Hecke characters.

The L-functions associated with Hecke's Gröβencharaktere, i.e. characters of infinite order, and their functional equations appear in a two-paper series beginning in 1918 [MR:1544392]. It is only in the second of these papers (1920) that Hecke uses the term "Gröβencharaktere".

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- Review status: reviewed
- Last edited by Stephan Ehlen on 2019-05-03 08:52:46

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- 2019-05-03 08:52:46 by Stephan Ehlen (Reviewed)
- 2019-05-03 08:52:11 by Stephan Ehlen
- 2016-05-09 21:54:18 by Brian Conrey

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