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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:

Early Years

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.