Are there any known analytic Artin L-Functions of 3 Dimensional representations?

Question

If so, what kind of underlying object(s) give rise to the analytic continuation of the Artin L-Function? Is it a cusp form or something else? To clarify, I mean how does one get the analytic continuation/functional equation of an Artin $L$-function attached to a 3-dimensional irreducible representation? Is there a function with properties, similar to modular forms and their action under $SL(2, \mathbb{Z})$, from which we can take the mellin transform of and give us said $L$-function with a $\Gamma$-factor that exhibits functional symmetry?