  The galaxy M 87 harbors a kiloparsec-scale relativistic jet, whose origin
coincides with a supermassive black hole. Observational mm-VLBI campaigns are
capable of resolving the jet-launching region at the scale of the event
horizon. In order to provide a context for interpreting these observations,
realistic general-relativistic magnetohydrodynamical (GRMHD) models of the
accretion flow are constructed. The characteristics of the observed
spectral-energy distribution (SED) depend on the shape of the electrons'
energy-distribution function (eDF). The dependency on the eDF is omitted in the
modeling of the first Event Horizon Telescope results. In this work, we aim to
model the M 87 SED from radio up to NIR/optical frequencies using a
thermal-relativistic Maxwell- J\"uttner distribution, as well as a relativistic
$\kappa$-distribution function. The electrons are injected based on sub-grid,
particle-in-cell parametrizations for sub-relativistic reconnection. A GRMHD
simulation in Cartesian-Kerr-Schild coordinates, using eight levels of adaptive
mesh refinement (AMR), forms the basis of our model. To obtain spectra and
images, the GRMHD data is post-processed with the ray-tracing code RAPTOR,
which is capable of ray tracing through AMR GRMHD simulation data. We obtain
radio spectra in both the thermal-jet and $\kappa$-jet models consistent with
radio observations. Additionally, the $\kappa$-jet models also recover the
NIR/optical emission. The models recover the observed source sizes and core
shifts and obtain a jet power of $\approx 10^{43}$ ergs/s. In the $\kappa$-jet
models, both the accretion rates and jet powers are approximately two times
lower than the thermal-jet model. The frequency cut-off observed at $\nu
\approx 10^{15}$ Hz is recovered when the accelerator size is $10^6$ - $10^8$
cm, this could potentially point to an upper limit for plasmoid sizes in the
jet of M 87.
