Levi has not had a precise birth year specified. We only know that he was in his early thirties in 850. Now, 4 years later, he is in his mid thirties.
He is certainly born before 820, btw. He was over 30 at the Battle of Trost, in the Spring. If we take Spring 850, and take away 30 years (Levi's minimum age), then he was over 0 years old in the Spring of 820. December 25th 820 is after Spring 820, therefore, Levi was already born.
If you don't have a source, bear in mind that head-canons aren't added to articles on any competent wiki.