So I was watching one of the videos out there explaining the game and in the basic weaknesses one of the enemies is the "Stubborn Detective". When he is at your location, treat your character card text box as blank. This means you do not have access to your special ability or your elder sign token ability.

My question is what happens then when you draw an elder sign token as a check while he is at your location? Does it simply function as a zero since you have no ability triggered by it, or something else?

It's a token that references nothing, so it doesn't give any modifiers, so treat it as 0.