Si-based photodetectors satisfy the criteria of low-cost and environmental-friendly, and can enable the development of on-chip complementary metal-oxide-semiconductor (CMOS)-compatible photonic systems. However, extending their room-temperature photoresponse into the mid-wavelength infrared (MWIR) regime remains challenging due to the intrinsic bandgap of Si. Here, we report on a comprehensive study of a room-temperature MWIR photodetector based on Si hyperdoped with Te. The demonstrated MWIR p-n photodiode exhibits a spectral photoresponse up to 5 {mu}m and a slightly lower detector performance than the commercial devices in the wavelength range of 1.0-1.9 {mu}m. We also investigate the correlation between the background noise and the sensitivity of the Te-hyperdoped Si photodiode, where the maximum room-temperature specific detectivity is found to be 3.2 x 10^12 cmHz^{1/2}W^{-1} and 9.2 x 10^8 cmHz^{1/2}W^{-1} at 1 {mu}m and 1.55 {mu}m, respectively. This work contributes to pave the way towards establishing a Si-based broadband infrared photonic system operating at room temperature.