Joint extreme values of L-functions

We consider L-functions L<inf>1</inf>, … , L<inf>k</inf> from the Selberg class which have polynomial Euler product and satisfy Selberg’s orthonormality condition. We show that on every vertical line s= σ+ it with σ∈ (1 / 2 , 1) , these L-functions simultaneously take large values of size exp(c(logt)1-σloglogt) inside a small neighborhood. Our method extends to σ= 1 unconditionally, and to σ= 1 / 2 on the generalized Riemann hypothesis. We also obtain similar joint omega results for arguments of the given L-functions.