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A monolithic X- and gamma-ray detector based on a silicon drift detector coupled to a CsI(Tl) scintillator is presented. The detector is operated both as a direct X-ray detector for photons interacting in silicon and as an indirect detector for photons interacting in the scintillator. As interactions in silicon and in CsI yield different amount of charge per unit energy deposited, discrimination of the place of interaction is necessary to obtain the correct energy. Discrimination of the interaction type is carried out by means of pulse shape discrimination performed with two parallel processing chains with different shaping times. In this paper an extensive characterization of the detector with temperature is presented. It is shown that cooling the detector at 10°C allows to obtain a nearly 100% efficiency between 8 and 200 keV. Further cooling below 0°C allows pulse shape discrimination with 100% accuracy throughout the detector's energy range. The detector has also been tested with X-rays at various energies by means of a tunable X-ray facility. These tests allowed an investigation of the light yield nonproportionality in CsI(Tl) at low energies, necessary for a proper energy calibration of the detector.