After losing its petition for a permit to build antennas on a building across from the Law School, AT&T filed a federal lawsuit on Friday against the City of Cambridge, stating that “[w]ireless service is important to public safety and convenience.”

According to the document filed at the U.S. District Court in Boston, AT&T sought to erect the antenna on the roof of 1558 Massachusetts Ave. in order to remedy the “significant gap” in its network coverage in the area north of Harvard Square.

The Cambridge Zoning Ordinance required AT&T to obtain a special permit in order to place the antenna on the residential building’s roof.

As a result, AT&T applied to the Cambridge Board of Zoning Appeals in December 2009. After AT&T revised their original application, the Cambridge Board of Zoning Appeals reviewed the wireless provider’s petition on June 24.

During the hearing, three of the five Board members voted in support of AT&T, but a vote of four is required to approve of an appeal.

The Board of Zoning Appeals cited in its decision that AT&T’s application failed to meet one or more of the five criteria under the Cambridge Zoning Ordinance’s section for special permits. The decision failed to specify which criteria were not met.

In its suit, AT&T claimed that the Board’s decision has caused “irreparable harm.” The company cited provisions of the Telecommunications Act, which seeks “to promote competition and reduce regulation in order to secure lower prices and higher quality services for American telecommunications consumers and encourage the rapid deployment of new telecommunications technologies.”

According to Constantine Alexander, chairman of the Board of Zoning Appeals, AT&T is not the first company to be turned down a special permit and file a lawsuit in Cambridge, though he could not recall other cases within the last six to eight years.

Installing the antennas could compromise the building’s aesthetics, Alexander said, suggesting a reason AT&T’s petition had been denied.

The document submitted on Friday explained that AT&T’s plans for facilities on top of the building on 1558 Massachusetts Ave. would consist of 12 antennas “mounted within six artificial (or faux) chimneys matching the existing chimneys,” and equipment cabinets in the basement of the building. At the hearing in June, the company also presented a sample of the fiberglass brick that would cover the chimneys.

Harvard Street Management, the company that manages the residential building in question, could not be reached for comment on Sunday.

—Staff writer Xi Yu can be reached at xyu@college.harvard.edu.