We report on the discovery of three transiting super-Earths around K2-155 (EPIC 210897587), a relatively bright early M dwarf ($V=12.81$ mag) observed during Campaign 13 of the NASA K2 mission. To characterize the system and validate the planet candidates, we conducted speckle imaging and high-dispersion optical spectroscopy, including radial velocity measurements. Based on the K2 light curve and the spectroscopic characterization of the host star, the planet sizes and orbital periods are $1.55_{-0.17}^{+0.20},R_oplus$ and $6.34365pm 0.00028$ days for the inner planet; $1.95_{-0.22}^{+0.27},R_oplus$ and $13.85402pm 0.00088$ days for the middle planet; and $1.64_{-0.17}^{+0.18},R_oplus$ and $40.6835pm 0.0031$ days for the outer planet. The outer planet (K2-155d) is near the habitable zone, with an insolation $1.67pm 0.38$ times that of the Earth. The planets radius falls within the range between that of smaller rocky planets and larger gas-rich planets. To assess the habitability of this planet, we present a series of 3D global climate simulations assuming that K2-155d is tidally locked and has an Earth-like composition and atmosphere. We find that the planet can maintain a moderate surface temperature if the insolation proves to be smaller than $sim 1.5$ times that of the Earth. Doppler mass measurements, transit spectroscopy, and other follow-up observations should be rewarding, since K2-155 is one of the optically brightest M dwarfs known to harbor transiting planets.
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