Hecke L-functions (reviewed)

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".