Self-reciprocal functions, powers of the Riemann zeta function and modular-type transformations

Integrals containing the first power of the Riemann Ξ-function as part of the integrand that lead to modular-type transformations have been previously studied by Ramanujan, Hardy, Koshlyakov, Ferrar and others. An integral containing the square of the Riemann Ξ-function and involving an extra parameter z, whose type naturally extends that of the afore-mentioned integrals, was studied by Ramanujan. This integral implicitly involves squaring of the functional equation of ζ(s). A unifying procedure to analyze general integrals of this type is studied here along with the interesting modular transformations that they generate. This also includes generalization of some transformations of Koshlyakov involving a series containing the modified Bessel function K0(x).