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

We show that for at least $$\frac{5}{13}$$ of the primitive Dirichlet characters $$\chi $$ large prime modulus, central value $$L(\frac{1}{2},\chi )$$ does not vanish, improving on previous best known result $$\frac{3}{8}$$ .