  Given a half-integral weight holomorphic Kohnen newform $f$ on $\Gamma_0(4)$,
we prove an asymptotic formula for large primes $p$ with power saving error
term for \begin{equation*} \sideset{}{^*} \sum_{\chi \hspace{-0.15cm} \pmod{p}}
| L(1/2,f,\chi) |^2. \end{equation*} Our result is unconditional, it does not
rely on the Ramanujan--Petersson conjecture for the form $f$. This gives a very
sharp Lindel\"{o}f on average result for Dirichlet series attached to Hecke
eigenforms without an Euler product. The Lindel\"{o}f hypothesis for such
series was originally conjectured by Hoffstein. There are two main inputs. The
first is a careful spectral analysis of a highly unbalanced shifted convolution
problem involving the Fourier coefficients of half-integral weight forms. The
second input is a bound for sums of products of Sali\'{e} sums in the
Polya--Vinogradov range. Half--integrality is fully exploited to establish such
an estimate. We use the closed form evaluation of the Sali\'{e} sum to relate
our problem to the sequence $\alpha n^2 \pmod{1}$. Our treatment of this
sequence is inspired by work of Rudnick--Sarnak and the second author on the
local spacings of $\alpha n^2$ modulo one.
