Faint millimeter NIKA2 dusty star-forming galaxies: finding the high-redshift population

We develop a new framework to constrain the source redshift. The method jointly accounts for the detection/non-detection of spectral lines and the prior information from the photometric redshift and total infrared luminosity from spectral energy distribution analysis. The method uses the estimated total infrared luminosity to predict the line fluxes at given redshifts and generates model spectra. The redshift-dependent spectral models are then compared with the observed spectra to find the redshift. Results. We apply the aforementioned joint redshift analysis method to four high-z dusty star-forming galaxy candidates selected from the NIKA2 observations of the HLSJ091828.6+514223 (HLS) field, and further observed by NOEMA with blind spectral scans. These sources only have SPIRE/Herschel photometry as ancillary data. They were selected because of very faint or no SPIRE counterparts, as to bias the sample towards the highest redshift candidates. The method finds the spectroscopic redshift of 4 in the 5 NOEMA-counterpart detected sources, with z>3. Based on these measurements, we derive the CO/[CI] lines and millimeter continuum fluxes from the NOEMA data and study their ISM and star-formation properties. We find cold dust temperatures in some of the HLS sources compared to the general population of sub-millimeter galaxies, which might be related to the bias introduced by the SPIRE-dropout selection. Our sources, but one, have short gas depletion time of a few hundred Myrs, which is typical among high-z sub-millimeter galaxies. The only exception shows a longer gas depletion time, up to a few Gyrs, comparable to that of main-sequence galaxies at the same redshift. Furthermore, we identify a possible over-density of dusty star-forming galaxies at z=5.2, traced by two sources in our sample, as well as the lensed galaxy HLSJ091828.6+514223. (abridged)