The context I am thinking of mainly is GL(2), and accordingly, the degree 2 Artin L-function. But comments about GL(n) in general are also welcome.


In light the local Langlands correspondence, what can we say about Artin's L-function? For example, does it follow that for a fixed automorphic representation $\pi$ and $L$-group representation $r$ one may equate local factors on the automorphic side $L\_v(s,\pi,r)$ (at unramified places $v$, at least) with the local factors on the Galois side $L\_v(s,\varphi)$ for some admissible homomorphism $\varphi$? And say, how far out of reach is Artin's conjecture in degree two?



GL(2) Local Langlands and Artin's L-function
I am not sure exactly of what is asked in this question, but I wish to correct what has been said in an answer because it is a little bit outdated. (Edited after Kevin's comment)


For $n=2$, Artin's conjecture had been solved in the 80's by Langlands and Tunnel for representations of $G\_{\mathbb Q}$ to $GL\_2(\mathbb C)$ which have solvable image in $PGL\_2(\mathbb C)$.
(That is to say, such representations are attached to automorphic forms --- modular forms of weight 1 in the odd case, and Maass forms in the even case --- and therefore their L-functions are entire.).


This was the situation 30 years ago. But since Wiles' proof of FLT, many things have moved, and we now know Artin's conjecture for *all representations of dimension $2$ that are* **odd**, because it is a consequence of Serre's conjecture, which has been proved by Khare and Wintenberger. Hence in the odd case the case of $A\_5$ (that is where the image of the rep. in $PGL\_2(\mathbb C)$ is isomorphic to $A\_5$) is now on at equal footing as the other, solvable, cases.


The only thing (but it is a big thing) that remains to be done for Artin's conjecture for $n=2$ is the case of even representations with projective image $A\_5$. This seems really hard, as the natural strategy is to prove that such representation are attached to Maass form, but Maass forms are much less well understood than their holomorphic cousins (that is traditional modular forms).


For larger $n$, even under oddness assumption, only some sporadic results are known. The method of Khare-Wintenberger seems hard to generalize, not only because
generalizing their induction steps suppose a lot of hard work (more than that has been done) but because for $n$ large it seems that the "starting step of the induction" is missing.


Staying for $n=2$, one can also consider Artin's conjecture for representation
of $G\_K$ for $K$ a number field. When $K$ is totally real, and the representation odd, Artin's conjecture has been announced by Pilloni and Stroh (and also independently in many cases by Kassaei, Sasaki, and Tianan).


