Zero-free regions for derivatives of the Selberg zeta-function

Let $Z(s)$ be the Selberg zeta-function associated with a compact Riemann surface. We prove that, for any positive integer $k$, there is a constant $t_0$ such that $Z^{(k)}(s)$ has no zeros in $\sigma<1/2$, $t>t_0$. Moreover, we show that the curve $Z(1/2+it)$ spirals in the clockwise direction for all sufficiently large $t$, in the sense that its curvature is negative.