To which branch of the government does the Library of Congress belong?

It isn't part of any branch. It was established by an act of congress in 1800 when president John Adams signed a bill providing for the transfer of the seat of government from Philadelphia to the new capital of Washington. The legislation described a reference library for congress only, containing "such books as may be necessary for the use of congress and for putting up suitable apartment for containing therein". Established with 5,000 appropriated by legislation the original library was in the new capital until 1814, when the invading British set fire to the capital and burned the small library. A month later Jefferson donated his library of 6,000 books as a replacement. His library was considered one of the finest in the United States. In January 1815 congress accepted his offer and gave him 23,950 for 6,487 books and the foundation for a national library was created.