Not under the Constitution.
But when a vote of no-confidence passes the House of Representatives--or a confidence vote is rejected--the prime minister must either dissolve the lower house within 10 days or resign along with his or her cabinet ministers.
The Japanese Constitution says that the prime minister be a member of the National Diet and be designated by Diet resolution after receiving a majority of votes cast in a Diet election. A new prime minister must be appointed by the Emperor, although this is a formality.
In case the cabinet resigns or when the Diet is convened following an election, the designation of a prime minister takes precedence over all other Diet matters.
Historically, the person named to become prime minister has generally been the leader of the biggest party in the lower house.
The longest serving prime minister in the past was Taro Katsura, who was elected three times between June 1901 and February 1913. He was in the position for seven years and 11 months. The shortest was Naruhiko Higashikuni, who served the post only for 54 days between August and October 1945.
Photo courtesy of Ministry of Foreign Affairs.