Cessna 402 Captain reported while in icing conditions the Captain's airspeed indicator indicated zero. After landing at destination airport; flight crew found the left pitot heat failed.

Narrative

Original flight crew plane went down for maintenance in ZZZ; so my First Officer (FO) and I were asked to operate a different flight. It was snowing showers in the area of ZZZ. We took off from ZZZ and accumulated ice on departure. It was light ice; however no issue. We were able to get on top of the clouds/icing condition at 10000 ft. Center cleared us down to 8000 ft. for the arrival into ZZZ1. Once reaching 8000 ft. we were in moderate icing conditions. I noticed my airspeed indicator start to decrease and noticed enough ice on boots to blow them. After blowing the boots the airspeed still continued to decrease and eventually went to zero. My FO requested for descent out of icing/IMC conditions so they cleared us to 3000 ft. I disconnected the autopilot and started my descent. I confirmed that the pitot heat was on and was using the second GPS for my approximated ground speed. We confirmed our current altitude with ATC and realized the only instrument that was not accurate was my airspeed. Luckily this plane had standby instruments on the copilot side and were working. Approach cleared us for the visual approach to XXR. About a mile final my airspeed started to come back (presumably the ice melted off from the pitot tube). Once we parked at the gate I tested the pitot heat and noticed that the left was cold and the right was warm. Pitot heat failed in IMC/icing conditions caused airspeed to drop to zero.