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Ibrahimovic scored the first goal in the new stadium after 20 minutes when he powered the ball into the roof of the net and then equalised after 77 minutes, volleying an unstoppable shot past Joe Hart.
He completed his hat-trick by firing low past Hart after 84 minutes to put Sweden back in front and capped his performance with an incredible angled left-footed overhead kick, an immediate contender for the goal of the season.
England cancelled out his first goal when Danny Welbeck scored after 35 minutes, and Steven Caulker, one of the six debutants, scored his first goal for England three minutes later after a superb free kick from Gerrard.
Caulker, Leon Osman and 17-year-old Raheem Sterling all started for the first time, while Ryan Shawcross, Carl Jenkinson and Wilfried Zaha came on for their first caps as substitutes as Roy Hodgson lost for the first time in 12 matches as England manager.