Swami Vivekananda was born on 12 January 1863. Does the time of birth is mentioned anywhere?
Yes, It's around 6:33 AM
According to Vivekananda A Biography By Swami Nikhilananda:
Swami Vivekananda, the great soul loved and revered in East and West alike as the rejuvenator of Hinduism in India and the preacher of its eternal truths abroad, was born at 6:33, a few minutes before sunrise, on Monday, January 12, 1863. It was the day of the great Hindu festival Makarasamkranti, when special worship is offered to the Ganga by millions of devotees. Thus the future Vivekananda first drew breath when the air above the sacred river not far from the house was reverberating with the prayers, worship, and religious music of thousands of Hindu men and women.
Another biography: SWAMI VIVEKANANDA - A Biography By His Eastern and Western Disciples ADVAITA ASHRAMA (April 1960) mention in the section:
II. BIRTH AND CHILDHOOD
The light of the world dawned for the first time upon the future Swami Vivekananda on Monday, January 12, 1863. It was the holy hour of dawn just six minutes before the sunrise. At the time of his birth the constellation Sagittarius was rising in the east, the moon was in the constellation Virgo, the planet Jupiter was in the eleventh house, and Saturn was in the tenth from that of his birth. It was the seventh day of the new moon in the month of Poush which is the ninth month of the Bengali year and as chance would have it. it was the day of Makara Sankranti, a great Hindu festival.
Swami Vivekananda alizaliwa tarehe 12 Januari 1863. Je, wakati wa kuzaliwa unatajwa mahali popote? "Kama inavyoelezwa na Vivekananda, ""Swami Vivekananda, roho kubwa iliyopendwa na kuheshimiwa Mashariki na Magharibi kama vile mrekebishaji wa Uhindu nchini India na mhubiri wa ukweli wake wa milele nje ya nchi, alizaliwa saa 6:33 asubuhi, dakika chache kabla ya jua kuchomoza, Jumatatu, Januari 12, 1863.""" Siku hiyo ni siku ya sherehe kubwa ya Kihindu ya Makarasamkranti, ambapo mamilioni ya waabudu humwabudu Ganga. Hivyo, Vivekananda wa wakati ujao alipopumua kwa mara ya kwanza, hewa juu ya mto mtakatifu, karibu na nyumba yake, ilikuwa ikisikika kwa sauti ya sala, ibada, na muziki wa kidini wa maelfu ya wanaume na wanawake Wahindu. "Swami Vivekananda, ""A Biography By His Eastern and Western Disciples"" (Audio ya kitabu cha Advaita Ashram, Aprili 1960) " "Kutoka kwa ""Nuru ya Dunia"" ilitokea kwa mara ya kwanza kwa Swami Vivekananda mnamo 12 Januari 1863." Ilikuwa saa takatifu ya mapambazuko dakika sita tu kabla ya jua kuchomoza. Wakati wa kuzaliwa kwake, nyota ya Sagittarius ilikuwa ikiongezeka mashariki, mwezi ulikuwa katika nyota ya Virgo, sayari ya Jupiter ilikuwa katika nyumba ya kumi na moja, na Saturn ilikuwa katika kumi kutoka kwa ile ya kuzaliwa kwake. Siku ya saba ya mwezi mpya katika mwezi wa Poush, mwezi wa tisa wa mwaka wa Kibengali, ni siku ya kwanza ya mwezi. Siku hiyo ilikuwa siku ya Makara Sankranti, sherehe kubwa ya Kihindu.