Let $h: \mathbb{R} \mapsto \mathbb{C}$ be a positive definite function, continuous at the origin. (In fact, $h$ is the Fourier transform of a finite measure). Define the oscillatory integral $$Q(t) := \int_0^\infty \frac{\text{Im} \{ h(u)^t\}}{u} du, \quad t > 0.$$ Are there non-trivial examples of $h$ that makes $Q$ independent of $t$?

Is there a known class of functions $\{ h \}$ that have this property?