  Let L(s) = L(s, \pi) be the standard L-function of a cuspidal representation
\pi of GL(m,A) where A denotes the ad\`eles of the field of rationals. We
consider the integral, on the real line Re(s)= 1/2, of the squared absolute
value of L(s)/s. In an earlier paper, partly with P. Sarnak (arxiv:2203.12475)
we obtained a universal lower bound on this integral, independently of m. In
this paper, for m fixed, we first obtain a universal lower bound for the
integral on an interval [-A logC, A log C] where C is the analytic conductor of
\pi; this bound is of order c(log C)^{-1/2} ; A, c are absolute positive
constants for m fixed. There is also an absolute lower bound on a shifted
interval [X-T, X+T] where T is of the order of log X. In the second part of the
paper, using the Mellin transform as in the previous paper, we estimate, for an
irreducible, non trivial Galois representation \rho of Gal(E/F), E and F being
number fields, the smallest norm of a prime ideal P of F at which \rho is
unramified and \rho(Frob) is non-trivial, Frob being a Frobenius at P.
