The closest to $0$ spectral number of the partial theta function

Abstract: The {\em spectrum} of the partial theta function $\theta :=\sum {j=0}^{\infty}q^{j(j+1)/2}x^j$ is the set of values of $q\in \mathbb{C}$, $0<|q|<1$, for which $\theta (q,.)$ has a multiple zero. We show that the only element of the spectrum belonging to the disk $\mathbb{D}{0.31}$ is $0.3092493386\ldots$.