  The broadband X-ray spectra of AGNs contains information about the nuclear
environment from Schwarzschild radii scales to distances of ~1 pc. In addition,
the average shape of the X-ray spectrum is an important input into X-ray
background synthesis models. Here, local (z \approx 0) AGN luminosity functions
(LFs) in five energy bands are used as a low-resolution, luminosity-dependent
X-ray spectrometer in order to constrain the average AGN X-ray spectrum between
0.5 and 200 keV. The 15-55 keV LF measured by Swift-BAT is assumed to be the
best determination of the local LF, and then a spectral model is varied to
determine the best fit to the 0.5-2 keV, 2-10 keV, 3-20 keV and 14-195 keV LFs.
The spectral model consists of a Gaussian distribution of power-laws with a
mean photon-index <\Gamma> and cutoff energy E_cut, as well as contributions
from distant and disc reflection. The reflection strength is parameterised by
varying the Fe abundance relative to solar, A_Fe, and requiring a specific Fe K
equivalent width (EW). In this way, the presence of the X-ray Baldwin effect
can be tested. The spectral model that best fits the four LFs has <\Gamma> =
1.85 \pm 0.15, E_cut=270^{+170}_{-80} keV, A_Fe=0.3^{+0.3}_{-0.15} (90% C.L.).
The sub-solar A_Fe indicates the presence of strong reflection given the
assumed Fe K EW. Indeed, parameterising the reflection strength with the R
parameter gives R=1.7^{+1.7}_{-0.85}. There is moderate evidence for no X-ray
Baldwin effect. A critical result of our procedure is that the shape of the
local 2-10 keV LF measured by HEAO-1 and MAXI is incompatible with the LFs
measured in the hard X-rays by Swift-BAT and RXTE. We therefore present a new
determination of the local 2-10 keV LF that is consistent with all other energy
bands.This new LF should be used to revise current measurements of the evolving
AGN LF in the 2-10 keV band.
