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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.

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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.

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    What are conditions a,b and c? Presumably that $\rho$ is irreducible, odd and something else? It would be helpful if you would post where you found this. – 2017-02-11
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    This looks like a form of the two dimensional Artin conjecture, which is still open in some cases, but is only known in full generality when $\rho$ is odd due to the work of Khare and Wintenberger on Serre's conjecture. The attribution to Weil and Langlands suggests you are looking at a specific case, possibly there case where the projective image of $\rho$ is $A_4$, which was, I think, proven by Langlands. Please could you clarify? – 2017-02-11
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    Ah! Condition c really is crucial here. Without condition c, this is still a theorem as of 2009, and is known as the Strong Artin Conjecture. Condition c is the assumption that the Artin conjecture is true for $\rho$ and all it's twists. With condition c, this is called the [converse theorem for $\mathrm{GL}(2)$](https://en.m.wikipedia.org/wiki/Converse_theorem). The original reference is the paper of Weil that is cited, and is in German. I've not seen the proof, but it may be something that you want to accept on trust for now. – 2017-02-11

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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.