for what $\nu$ does Riemann-Liouville differintegral of digamma function $\psi(z)$ exist?

Question

For what values of $\nu$ does the Riemann-Liouville differintegral $_{-\infty}D_{z}^\nu$ of the digamma function $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ exist, with $c=-\infty$? All I've got so far is that the derivatives exist, i.e. the differintegrals when $\mathrm{Re}(\nu)>0 $.

Many thanks for any help with this!