An Artin representation is a continuous homomorphism ρ: Gal(ℚ/ℚ)→GL(V) from the absolute Galois group of ℚ to a finite dimensional complex vector space V. By continuity, this factors through the Galois group of some finite extension K/ℚ. The Euler factors Lp can be computed by taking completions of K, while the computations for the places of ramification are more difficult.
The main Magma function takes a number field K and computes all irreducible Artin representations that factor through the normal closure F of K. Various properties of the representations can be obtained. These include the conductor, kernel and ramification at a prime. Most importantly the local polynomial (Euler factor) of an Artin representation can be computed at a prime p.
Through the use of representation theory, the Dedekind ζ-function of K can then be decomposed into the corresponding Artin L-functions. As the Artin L-series have smaller conductors, it is easier to compute with them. By way of contrast, PARI/GP has to work directly with the Dedekind ζ-function. In this sense, the Artin L-series are the real primitive building blocks for number field computations, rather than the Dedekind ζ-functions.
The Magma implementation of Artin representations was originally developed by T. Dokchitser and then later expanded by Tim and Vladimir Dokchitser, and it relies heavily on the permutation group machinery available in Magma for identifying Frobenius elements in Galois groups. For instance, they have an example where the Galois group PGSp4(GF(3)) has order 51840 (coming from the action on 3-torsion of a genus 2 Jacobian), where the Dokchitser brothers can compute various special L-values to 10 digits in a few hours.