A cough lasting 15 to 20 days without fever is usually what's called a subacute cough, and the common causes are a lingering post viral cough, allergies, or reflux irritating your throat. It is not an emergency, but it has gone on long enough to need a proper look.
Next Steps
Start with a general physician rather than picking a specialist yourself. They can check for the common causes and order a chest X-ray if needed, then refer you to a pulmonologist or ENT only if something specific needs that expert's input.
Health Tips
See a doctor sooner if you develop fever, breathlessness, tightness in your chest, or notice blood when you cough.