  This Letter presents a search for the production of a long-lived neutral
particle ($Z_d$) decaying within the ATLAS hadronic calorimeter, in association
with a Standard Model (SM) $Z$ boson produced via an intermediate scalar boson,
where $Z\to l^+l^-$ ($l=e,\mu$). The data used were collected by the ATLAS
detector during 2015 and 2016 $pp$ collisions with a center-of-mass energy of
$\sqrt{s} = 13$ TeV at the Large Hadron Collider and corresponds to an
integrated luminosity of $36.1\pm0.8$ fb$^{-1}$. No significant excess of
events is observed above the expected background. Limits on the production
cross section of the scalar boson times its decay branching fraction into the
long-lived neutral particle are derived as a function of the mass of the
intermediate scalar boson, the mass of the long-lived neutral particle, and its
$c\tau$ from a few centimeters to one hundred meters. In the case that the
intermediate scalar boson is the SM Higgs boson, its decay branching fraction
to a long-lived neutral particle with a $c\tau$ approximately between 0.1 m and
7 m is excluded with a 95% confidence level up to 10% for $m_{Z_d}$ between 5
and 15 GeV.
