I have a random variable whose characteristic function is of the form
\begin{equation} \mathbb{E}[e^{itX}] = \frac{(1-it)^a}{(1-2it)^{\frac{a}{2}}}\,, \end{equation} where $0<a<1$
I am not even sure if this is indeed positive definite, and hence, a characteristic function. If we substitute $e^x \approx (1-x) $, we get $\mathbb{E}[e^{itX}] = 1$, which is the characteristic function of $\delta_0$.
Hence, prima facie, it appears to be the characteristic function of a distribution with most of its mass near 0.
Note that, the characteristic function of a gamma random variable is $(1-it)^{-a}$.
Any information about the distribution will be helpful.