  We consider the 1D nonlinear Schr\"odinger equation (NLS) with focusing
\emph{point nonlinearity}, $$i\partial_t\psi + \partial_x^2\psi +
\delta|\psi|^{p-1}\psi = 0$$ where $\delta=\delta(x)$ is the delta function
supported at the origin. In the $L^2$ supercritical setting $p>3$, we construct
self-similar blow-up solutions belonging to the energy space $L_x^\infty \cap
\dot H_x^1$. This is reduced to finding outgoing solutions of a certain
stationary profile equation. All outgoing solutions to the profile equation are
obtained by using parabolic Weber functions and solving the jump condition at
$x=0$ imposed by the $\delta$ term. This jump condition is an algebraic
condition involving gamma functions, and existence and uniqueness of solutions
is obtained using the intermediate value theorem and formulae for the digamma
function. We also compute the form of these outgoing solutions in the slightly
supercritical case $0<p-3 \ll 1$ using the log Binet formula for the gamma
function, and contour deformation and stationary phase/Laplace method in the
integral formulae for the parabolic Weber functions.
