Before you send it in for service you might want to try a few things.
First, wear the watch for a month or so. The movement should break-in and accuracy improve.
Secondly, since the watch is running slow, it might not have enough power. Try winding the watch by hand for a few days. It will add some wear to the crown to wind it but otherwise doesn't hurt an automatic to wind by hand. If this helps, you'll know you aren't active enough to keep the watch fully wound. (This is a common problem for people who work in offices.)
Thirdly, watches run slower or faster in different positions. Try changing the position of the watch when you take it off for the night. For example, leave it crown down for a few days, then crown up for a few day, then flat on its back for a few days, etc. You don't have to go through the whole cycle. When you find a position that works for you, just adopt it.
Also, for a mechanical watch to be a COSC certified chronometer it must not lose more than 4 nor gain more than 6 seconds per day. Most watches are not certified. Some are not certified because of the certification cost but most watches can't meet the accuracy requirement. Since your watch will never be bang on, you should consider how much time and money you want to spend chasing a few seconds a day.
You didn't mention the specific watch so this advice is fairly general.