Contradiction with Matthew 2:1
While Luke 2:15 talks about shepherds visiting Jesus on the night of his birth, Matthew 2:1 describes wise men (Magi) arriving from the east, implying a later time after Jesus' birth.
Matthew 2:1: Now when Jesus was born in Bethlehem of Judaea in the days of Herod the king, behold, there came wise men from the east to Jerusalem,
Contradiction with Matthew 2:11
Luke 2:15 details the shepherds visiting Jesus in a manger, whereas Matthew 2:11 states that the wise men found Jesus in a house.
Matthew 2:11: And when they were come into the house, they saw the young child with Mary his mother, and fell down, and worshipped him: and when they had opened their treasures, they presented unto him gifts; gold, and frankincense, and myrrh. [presented: or, offered]