  We present the average rest-frame spectrum of high-redshift dusty,
star-forming galaxies from 250-770GHz. This spectrum was constructed by
stacking ALMA 3mm spectra of 22 such sources discovered by the South Pole
Telescope and spanning z=2.0-5.7. In addition to multiple bright spectral
features of 12CO, [CI], and H2O, we also detect several faint transitions of
13CO, HCN, HNC, HCO+, and CN, and use the observed line strengths to
characterize the typical properties of the interstellar medium of these
high-redshift starburst galaxies. We find that the 13CO brightness in these
objects is comparable to that of the only other z>2 star-forming galaxy in
which 13CO has been observed. We show that the emission from the high-critical
density molecules HCN, HNC, HCO+, and CN is consistent with a warm, dense
medium with T_kin ~ 55K and n_H2 >~ 10^5.5 cm^-3. High molecular hydrogen
densities are required to reproduce the observed line ratios, and we
demonstrate that alternatives to purely collisional excitation are unlikely to
be significant for the bulk of these systems. We quantify the average emission
from several species with no individually detected transitions, and find
emission from the hydride CH and the linear molecule CCH for the first time at
high redshift, indicating that these molecules may be powerful probes of
interstellar chemistry in high-redshift systems. These observations represent
the first constraints on many molecular species with rest-frame transitions
from 0.4-1.2mm in star-forming systems at high redshift, and will be invaluable
in making effective use of ALMA in full science operations.
