After 13 years and three championships, the Miami Heat says its home court finally has generated enough profit to pay rent to its landlord, Miami-Dade County. On Friday, a Heat affiliate delivered a check for $257,134.12 to Miami-Dade that represents the public’s portion of a revenue-sharing agreement from a season that generated about $30 million in cash profits for the county-owned AmericanAirlines Arena. The infusion of money will make a tiny dent in the roughly $6.5 million operating subsidy the arena receives each year from the county. But the payment does mark a milestone in a financing arrangement where packed stands and high ticket sales had still allowed the Heat to occupy an arena rent-free. The Miami Heat organization funded construction of the waterfront facility on county-owned land, in exchange for the yearly operating subsidy and an agreement to share profits with Miami-Dade once the arena generates more than $14 million a year. But despite packed stands during Shaquille O’Neal’s four seasons with the team and even better ticket sales with the arrival of LeBron James in 2010, the Heat has always reported arena financials that fell short of requiring any payment to Miami-Dade. “Of course, we are very happy to receive the $257,314 check,” Mayor Carlos Gimenez said in a statement Friday. Sammy Schulman, the Heat’s chief financial officer, declined an interview request Friday, citing the team’s turkey giveaway to the needy that day. He said the team would answer questions on the financial reports Monday. Miami’s fickle relationship with the Heat helps explain the franchise’s rent-free years. The arena reported losing money in the years without Shaq or James, and the team’s agreement with Miami-Dade allows it to deduct past losses from current profits. When James joined Dwyane Wade and Chris Bosh to form the celebrated “Three Kings,” the surge in arena revenue began eating away at the deficit, and the outstanding balance vanished last year. The debut payment Friday on a 13-year-old arrangement comes as the Heat is trying to extend its current deal with Miami-Dade under terms that could include a higher subsidy. In June, team executives warned the arena needs a significant renovation to remain the kind of top-notch facility that forms a key economic hub in Miami’s emerging downtown entertainment scene. A Heat-funded economic impact study contends the arena contributes about $1.5 billion a year to the local economy. Team lobbyist Jorge Lopez said at the time the team may require a subsidy as high as $17 million a year to finance the kind of renovations the Heat says it needs for the government-owned facility. While two separate businesses, the entity that runs the arena and the entity that runs the team are both under the corporate umbrella of majority owner Micky Arison, the billionaire chairman of Carnival Corp. and one of the wealthiest people in the United States. The Heat’s current agreement, which uses hotel taxes to fund the subsidy, expires in 2029, but the Heat this year exercised its option to begin negotiating an extension. Heat executives said the timing was not related to the possibility of losing James to free agency after this season. The agreement with the county calculates cash profits (really, net cash flow) as the difference between most operating revenue and expenses, but the costs do not include expenses tied to arena financing and other categories. A separate financial statement for the arena lists the arena’s final profit at $17 million for 2013.