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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?

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    This question suggests a certain level of basic confusion. At the very least "underlying objects" is so vague as to not mean much. What are you imagining to be a satisfactory answer for 2-dimensional representations? Or 1-dimensional representations for that matter? – 2017-02-09
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    @ThePotter'sVessel The only basic confusion I have is about where Artin L-Functions of 3-Dimensional reps. get their functional equation from. 1-D reps get their functional equation from half integral weight cusp forms, the 2-D reps Ive had the chance to come across all got their functional equations from weight 1 cusp forms. Im curious to know where 3-D reps come from in terms of their functional equation. – 2017-02-09
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    When you say "the only basic confusion I have" you are merely indicating that you don't know enough to understand what you don't know. There is a procedure (conjectural in general) for going from an automorphic representation $\pi$ (together with some auxiliary info. such as a finite dimensional representation of the underlying group) to an $L$-series $L(\pi,s)$. This map is far from being injective, so there are "many" sources $\pi$ for the same $L$-function. Only in exceptional circumstances do any of these functions/representations have anything to do with modular forms (even for $n=2$). – 2017-02-09
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    @ThePotter'sVessel Well that settles it. Okay thats cool, I didn't know it was exceptional to find cusp forms for 2 dimensional representations... I mean I'm not in the position where I can pick a representation and then get a cusp form from it, its more the other way around. It looks like I've been living under a rock.... But from your response I've got so many more questions that it may be better to find sources to work up to this – 2017-02-09
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    I would say $\prod_{i=1}^n L(s,\chi_i)= L(s,\rho)$ for some n-dimensional representation $\rho$ of the Galois group of $L/K$, where the $\chi_i$ are Dirichlet characters of $K = \mathbb{Q}$ (or Hecke characters of the same number field $K$) and $L$ is a compositum field. If they are characters of different fields then $\rho$ is $\sum_{i=1}^n [F_i:K]$ dimensional where $K$ is a common subfield. @ThePotter'sVessel – 2017-10-06

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