3rd Regiment, Virginia Artillery (Confederate)
This unit is not in the Civil War Soldiers and Sailors Database. However it is in the Confederate Regimental Information, Virginia, Artillery, Cavalry, Infantry Regimental Histories, 3rd Artillery Regiment. 
Companies in this Regiment with the Counties of Origin
Men often enlisted in a company recruited in the counties where they lived though not always. After many battles, companies might be combined because so many men were killed or wounded. However if you are unsure which company your ancestor was in, try the company recruited in his county first.
National Park Service, The Civil War Soldiers and Sailors System, is searchable by soldier's name and state. It contains basic facts about soldiers on both sides of the Civil War, a list of regiments, descriptions of significant battles, sources of the information, and suggestions for where to find additional information.