If a character were suspicious about a door and wanted to examine it for signs of protection, magical or otherwise, I'd probably ask them to make an Intelligence check, let them know the suitable DC, and let them know the stakes (e.g. if you succeed, I'll tell you if you found anything; if you fail, the examination will go awry). And so, yes, a successful INT (Investigation) check here would indeed reveal that somehow this door is magically warded; a successful INT (Arcana) check might do that and perhaps clue them in to the spell being used.
Detect Magic would yield the result that some abjuration magic is on the door.
As an ongoing spell effect, it's detectable by detect magic.
As there is no effect to perceive, there is no perception check to notice it, however other signs might reveal something is suspicious about the warded area (no footprints into or out of the warded area, no lock despite all other doors in the area being locked etc). That's really a judgement call.