well several questions would have to be answered, like is is legal to discharge a AIR gun in your zoning /area!
next would be what are the legal hunting rules on shooting pigeons, it varies from state to state, so you need to check with your states game dept, they would also then be able to tell you how to dispose of them too, but typically you cannot just kill anything for no real reason, and can get some serious fines for doing so if NOT legal, as well as again discharging a air rifle in a area where its illegal to do so!