Seagate are really bad these days. Two of my customers have had them fail, exactly as you describe, within their 2 Year Warranty. Unless you bought their additional cost, Data protection, they will want hndreds of pounds to even attempt to recover the data.
Warning! Do not open the Outer case yet! that will invalidate any Warranty in their eyes although that is questionable in UK and EU Law.
Sorry to sound pessimistic. Do as described above in a silent environment. Place a finger lightly on the HDD as you plug it in and listen for "Clicks" which are a giveaway for a failed disc. You may find it doesn't spin up.
Post back with the results. I have had several dealings with them.