Joint extreme values of L-functions

We consider L -functions L 1 , … , L k from the Selberg class which have polynomial Euler product and satisfy Selberg’s orthonormality condition. We show that on every vertical line s = σ + i t with σ ∈ ( 1 / 2 , 1 ) , these L -functions simultaneously take large values of size exp c ( log t ) 1 - σ log log t 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.