Title: The strange consequences of Siegel zeros

Abstract: If you believe the Generalised Riemann Hypothesis, then there are no zeros of L-functions with real part bigger than 1/2, but unfortunately we don't know how to show this. A `Siegel zero' is a putative strong counterexample to GRH, and if such exceptional zeros do exist, then there are many strange consequences for the distribution of prime numbers. However, prime numbers would also become very regular, and this allows us to prove things which go beyond even the consequences of GRH, if these exceptional zeros exist! I will survey some of these results, including recent joint work showing we can prove results towards the horizontal Sato-Tate conjecture for Kloosterman sums in this alternative world where Siegel zeros exist.