Joint Extreme values of $L$-functions

Abstract

We consider $L$-functions $L_1,\dots ,L_k$ from the Selberg class which have polynomial Euler product and satisfy Selberg's orthonormality condition. We show that on every vertical line $s=\sigma +it$ with $\sigma \in (1/2,1)$, these $L$-functions simultaneously take large values of size $\exp \left(c\frac{(\log t{)}^{1-\sigma }}{\log \log t}\right)$ inside a small neighborhood.