Could anyone please provide a reference for a proof of this theorem? It was mentioned on p48 of this article without proof and I couldn't find a proof of it anywhere.
The conditions in the theorem are
a) $\rho$ is odd
b) $\rho$ is irreducible
c) For all continuous characters $\chi:G_\mathbb Q\to \mathbb C^\times$, the Artin L-function $L(\rho\otimes\chi,s)$ has analytic continuation to the entire complex plane.
As noted by Mathmo123 in comments, this is a form of the converse theorem.
There is the original paper of Weil in which he proved this (but I'm not sure if he treated the weight 1 case which is relevant to the Artin conjecture; I think he was more focussed on the weight 2 case, which relates to the modularity conjecture for elliptic curves).
The relationship b/w the converse theorem and the Artin conjecture was first observed by Langlands (as far as I know), and there is the famous book of Jacquet and Langlands, where they prove a form of the converse theorem (using adelic language, and working over arbitrary number fields).
To get a feeling for this subject, you could look at the old book of Ogg called (if I am remembering correctly) Dirichlet series and automorphic forms, which discusses the converse theory in classical terms, starting with Riemann's second proof of the functional equation of the $\zeta$-function and Hecke's theory, and ending (I think) with Weil's theorem.
You could also look at the article of Serre from a conference proceedings (I think the Durham proceedings from 1974), in which he discusses the relationship between Artin's conjecture and weight one forms, with examples.