Secant zeta functions

We study the series ψs(z)≔∑n=1∞sec(nπz)n−s and prove that it converges under mild restrictions on zz and ss. The function possesses a modular transformation property, which allows us to evaluate ψs(z)ψs(z) explicitly at certain quadratic irrational values of zz. This supports our conjecture that π−kψk(j)∈Q whenever kk and jj are positive integers with kk even. We conclude with some speculations on the Bernoulli numbers.