In a previous paper, we proposed a new method to select low-power BL Lacs (LPBLs) based on mid-infrared emission and flux contrast through the Ca II spectral break; that study led to the selection of a complete sample formed by 34 LPBLs with 0.05<z<=0.15 and radio luminosities spanning the range log(L_r) = 39.2-41.5 [erg/s]. We now assemble the broadband spectral energy distributions (SEDs) of these sources to investigate their nature and compare them with brighter BL Lacs. We find that the ratios between the X-ray and radio luminosities range from ~20 to ~30000 and that the synchrotron peak frequencies span a wide energy interval, from log(nu_peak)~13.5 to ~20 [Hz]. This indicates a broad variety of SED shapes and a mixture of BL Lac flavors. Indeed, although the majority of our LPBLs are high-energy peaked BL Lacs (HBLs), we find that a quarter of them are low-energy peaked BL Lacs (LBLs), despite the fact that the sample is biased against the selection of LBLs. The analysis of the median LPBL SED confirms disagreement with the blazar sequence at low radio luminosities. Furthermore, if we limit the sample to the LBLs subsample, we find that their median SED shape is essentially indistinguishable from that of the most luminous BL Lacs. We conclude that the observed radio power is not the main driving parameter of the multiwavelength properties of BL Lacs.
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