Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Ueber die Zetafunktionen gewisser algebraischer Zahlkorper, Math. Ann. 89, pp. 147-156]. This of course is a very special case of the general Artin holomorphy conjecture, and it is easy in the case of a normal extension. Dedekind did the case of pure cubic fields, but doesn't this result of his date to much later than 1873? Artin quotes the following paper from 1900, and attributes to it the result on pure cubic fields: [Dedekind, Ueber die Anzahl von Idealklassen in reinen kubischen Zahlkorpern, the Crelle journal, vol. 121, pp. 40-123].

Regarding the first of these conjectures, I believe it was first explicitly stated (in the more general setting of a relative extension $K/k$) in Artin's 1923 paper [Über die Zetafunktionen gewisser algebraischer Zahlkörper, Math. Ann. 89, pp. 147-156]. This of course is a very special case of the general Artin holomorphy conjecture, and it is easy in the case of a normal extension. Dedekind did the case of pure cubic fields, but doesn't this result of his date to much later than 1873? Artin quotes the following paper from 1900, and attributes to it the result on pure cubic fields: [Dedekind, Über die Anzahl von Idealklassen in reinen kubischen Zahlkörpern, the Crelle journal, vol. 121, pp. 40-123].