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.

Number-theory and algebraic geometry, Proc. Int. Cong. Math. 1950, vol. 2, pp. 90-100, which ends: "I should like to conclude with a brief discussion of a very interesting conjecture, due, I believe, to Hasse (...) we are thus led to considerthe product of these zeta-functions for all$\mathfrak p$, which is precisely the function previously defined by Hasse, of which he conjectured that it can be continued analytically over the whole plane, that it is meromorphic, and that it satisfies a functional equation."? $\endgroup$1more comment