Abstract: The Riemann zeta function has a meromorphic continuation to the whole complex plane with a simple pole at s=1 and no other poles. In other words, the zeta function has an analytic continuation to C apart from wellunderstood poles. Similar results are true for the Lfunctions attached to Dirichlet characters and, more generally, Hecke characters. Emil Artin reformulated these results as saying that the Lfunctions associated to 1dimensional complex Galois representations had analytic continuation apart from wellunderstood poles and conjectured that the same should be true for ndimensional complex Galois representations. This conjecture could now be regarded as a precursor to the Langlands programme. I will explain the embarassingly small number of positive results we know about Artin's conjecture. This talk will be for nonexperts; I will spend about half the talk defining complex Galois representations and their Lfunctions, and the other half giving statements of results and sometimes indications of proofs.
