Superimposing theta structure on a generalized modular relation

A generalized modular relation of the form $$F(z, w, \alpha )=F(z, iw,\beta )$$ , where $$\alpha \beta =1$$ and $$i=\sqrt{-1}$$ is obtained in course evaluating an integral involving Riemann $$\Xi $$ -function. This involves a surprising new generalization Hurwitz zeta function $$\zeta (s, a)$$ which we denote by _w(s, . We show that satisfies beautiful theory generalizing In particular, it shown for $$0-1$$ except simple pole at $$s=1$$ The theories functions reciprocal kernel combination Bessel modified $${}_1K_{z,w}(x)$$ are also essential obtain relation, developed.