The following question arises while I'm reading a paper of Jerzy Kaczorowski and Alberto Perelli (A correspondence theorem for L-functions and partial differential operators, Publ. Math. Debrecen 79/3-4 (2011), 497-505).

Consider an eigenfunction $f$ of hyperbolic laplacian $\Delta=-y^2(\frac{\partial ^2}{\partial x^2}+\frac{\partial ^2}{\partial y^2})$ defined on the upper half-plane $\mathbb{H}$(eigenvalue is real).$f$ satisfies the following conditions:

1)$f(z)=f(z+1),z=x+iy$;

2)$f(z)=o(e^{2\pi y}),y\rightarrow+\infty;$

3)If $z=iy,y>0$,then $z^pf(z)=Kf(-\frac{1}{qz}),p\geq 0,q>0,K\in\mathbb{C}$.

Question:Is it possible that 3) is true for every $z\in\mathbb{H}$?