A generalization of the Beurling–Lax theorem

We obtain conditions for the completeness of the system $\{G(z)e^{\tau z},\tau\leqslant0\}$ in the space $H^2_\sigma(\mathbb C_+)$, $0<\sigma<+\infty$, of functions analytic in the right-hand half-plane for which
$$ \|f\|:=\sup_{-\pi/2<\varphi<\pi/2}\biggl\{\,\int_0^{+\infty}|f(re^{i\varphi})|^2e^{-2r\sigma|\sin\varphi|}\,dr\biggr\}^{1/2}<+\infty. $$