Context. Wide-field Spitzer surveys allow identification of thousands of potentially high-z submillimeter galaxies (SMGs) through their bright 24 μm emission and their mid-IR colors. Aims. We want to determine the average properties of such z ∼ 2 Spitzer-selected SMGs by combining millimeter, radio, and infrared photometry for a representative IR-flux (λ rest ∼ 8 μm) limited sample of SMG candidates. Methods. A complete sample of 33 sources believed to be starbursts ("5.8 μm-peakers") was selected in the (0.5 deg 2 ) J1046+56 field with selection criteria F 24 μm > 400 μJy, the presence of a redshifted stellar emission peak at 5.8 μm, and r Vega > 23. The field, part of the SWIRE Lockman Hole field, benefits from very deep VLA/GMRT 20 cm, 50 cm, and 90 cm radio data (all 33 sources are detected at 50 cm), and deep 160 μm and 70 μm Spitzer data. The 33 sources, with photometric redshifts ∼1.5−2.5, were observed at 1.2 mm with IRAM-30m/MAMBO to an rms ∼0.7-0.8 mJy in most cases. Their millimeter, radio, 7-band Spitzer, and near-IR properties were jointly analyzed.Results. The entire sample of 33 sources has an average 1.2 mm flux density of 1.56 ± 0.22 mJy and a median of 1.61 mJy, so the majority of the sources can be considered SMGs. Four sources have confirmed 4σ detections, and nine were tentatively detected at the 3σ level. Because of its 24 μm selection, our sample shows systematically lower F 1.2 mm /F 24 μm flux ratios than classical SMGs, probably because of enhanced PAH emission. A median FIR SED was built by stacking images at the positions of 21 sources in the region of deepest Spitzer coverage. Its parameters are T dust = 37 ± 8 K, L FIR = 2.5 × 10 12 L , and SFR = 450 M yr −1 . The FIR-radio correlation provides another estimate of L FIR for each source, with an average value of 4.1 × 10 12 L ; however, this value may be overestimated because of some AGN contribution. Most of our targets are also luminous star-forming BzK galaxies which constitute a significant fraction of weak SMGs at 1.7 z 2.3. Conclusions. Spitzer 24 μm-selected starbursts and AGN-dominated ULIRGs can be reliably distinguished using IRAC-24 μm SEDs. Such "5.8 μm-peakers" with F 24 μm > 400 μJy have L FIR 10 12 L . They are thus z ∼ 2 ULIRGs, and the majority may be considered SMGs. However, they have systematically lower 1.2 mm/24 μm flux density ratios than classical SMGs, warmer dust, comparable or lower IR/mm luminosities, and higher stellar masses. About 2000−3000 "5.8 μm-peakers" may be easily identifiable within SWIRE catalogues over 49 deg 2 .