TITLE: First formulation of the Dedekind and Hasse-Weil conjectures
QUESTION [15 upvotes]: I'm looking for the original statement of two important conjectures in number theory concerning L-functions. I'm particularly interested in pinning down the year in which they were first formulated:

The Dedekind conjecture (that the quotient $\zeta_K(s)/\zeta(s)$ is entire for any number field $K$)

The only information I've been able to find is that Dedekind himself proved the conjecture in the case of pure cubic fields in 1873. This definitely gives an upper bound. I'm also curious to know for which extensions $K$ over $\mathbb{Q}$ he expected the conjecture to be true.
Edit (1) I've found the paper in question:
Richard Dedekind, Über die Anzahl von Idealklassen in reinen kubischen Zahlkörpern (1900)
Here is a link to his complete works. I can't read german, so the question is still the same, but I thought the link might help. Discussion about the zeta function of number fields seems to start in page 166 (section 6). 
Edit (2) Another clue, mentioned by Vesselin Dimitrov in the comments, and suggesting that the conjecture might be due to Landau:
"On page 34 of Bombieri's article The classical theory of zeta and L-functions in the Milan Journal of Mathematics, the conjecture is attributed to Landau. So it probably appears in Landau's Handbuch, much earlier than in Artin"

The Hasse-Weil conjecture (the zeta function of an algebraic variety has a meromorphic continuation to the complex plane and a functional equation)

(note: this has been nicely answered by Francois Ziegler in the comments)
The problem with this one seems to be that it applies to several different kinds of functions, and both definitions and result came in several stages. I think that the first and most basic definitions is Artin's zeta function of a curve over a finite field. This was generalized to arbitrary algebraic varieties, and also to schemes of finite type over $\mathbb{Z}$. Yet the most common definition is over number fields. There's also the local/global and L/zeta distinction.
In this case I'm just trying to get a feeling of when/where the conjecture was first formulated in some generality.

REPLY [5 votes]: Artin (1923) wrote that Dedekind proved the case of this conjecture concerning pure cubic number fields. Dedekind published his article in 1900, but writes that it is a reworking of a draft he had written in 1871/72. The conjecture was named after Dedekind by van der Waall in 1974. 
Despite Artin's claim, the integrality of the quotient $\zeta_K(s)/\zeta(s)$ is not addressed at all by Dedekind (Hecke proved that $\zeta_K(s)$ extends to a meromorphic function on the complex plane in 1917). Dedekind proved that  $\zeta_K(s)/\zeta(s)$ for pure cubic extensions may be written as a linear combination of Epstein zeta functions. Epstein proved in 1903 that these are meromorphic with simple poles in $s = 1$, and Artin realized in 1923 that this implies that the quotient is entire - I guess this explains his remark. 
In any case Dedekind had nothing to do with this conjecture.
Edit. Landau [Über die Wurzeln der Zetafunktion eines algebraischen Zahlkörpers, 
Math. Ann. 79 (1919), 388-401] writes on p. 390: "I have to be careful with its formulation since it is not known whether $\zeta_k(s)/\zeta(s)$ is always an entire function".