Added, 25 July:
Having thought about it a bit more, it occurs to me that this is yet another situation where Langlands urges us to go beyond a classical framework in seeking answers to non-abelian questions. For example, when we associate to an odd Artin represention
of dimension two an Artin $L$-function we can perfectly sensibly assert that the $a_n$'s follows a pattern. When asked what that pattern is, the answer, satisfying to some and mysterious to others, is that$$\sum a_nq^n$$is a modular form. This is the kind of thing that comes out of Langlands.
Now, if we want to 'characterize,' say, odd $GL_2$ Galoisian sets, we can say the following: Enumerate the normalized holomorphic Hecke (new) eigenforms $f$ of weight one sorted by level $N$ and character $\epsilon$.For each such form, run over the prime numbers $p$ not dividing $N$, and take the number $a_p$ defined by the equationfor the $p$-th Hecke operator $T_p$. Now look at the set$S_f$ of primes $p$ such that $$(p,N)=1, \epsilon (p)=1, a_p=2.$$These $S_f$'s are exactly the odd $GL_2$ Galoisian sets.
Perhaps it's unreasonable to want more from the Langlands' programme. Whether or not this is the final word on all such questions, well, that's a different matter.