Artin's conjecture on Artin $L$-functions asserts that the Artin $L$-function $L(\rho,s)$ of a non-trivial irreducible representation $\rho$ of the Galois group of a number field admits analytic continuation to the whole complex plane. 

It is known for $1$-dimensional and induced representations, plus a few other special cases.

What is the status towards a proof? References would be very much appreciated.

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Langlands, R. P. (1970), "Problems in the theory of automorphic forms", Lectures in modern analysis and applications, III, Lecture Notes in Math 170, Berlin, New York: Springer-Verlag, pp. 18–61.