Nero's birth day is listed in Suetonius, The Lives of Twelve Caesars, Life of Nero 6. His death day is uncertain, though, perhaps because Galba was declared emperor before Nero died. A June 9th death day comes from Jerome, Chronicle, which lists Nero's rule as 13 years, 7 months and 28 days. Cassius Dio, Roman History LXII.3 and Josephus, War of the Jews IV, say Nero's rule was 13 years, 8 months which would be June 11.