Did your doctor have you obtain a sputum sample? I say this because antibiotics will not cure a viral infection, they only work for bacterial infections. It's also important to know that most colds are not cause by bacteria, yet many doctors treat all colds (bronchitic, etc) with antibiotics. The only way to know for sure you have a bacteria is to obtain a sputum specimin and to run some tests on it.
A z-pack is a broad spectrim antibiotic. It's supposed to kill a wide variety of bacteria. If you do have a bacteria, and the z-pack didn't work, that might also mean you need a different antibiotic. A sputum specimin can not only tell a doctor if you have a bacteria, it also figures out what bacteria and what antibiotic kills it.
It's also important to note that sometimes if a patient has a chronic disease, such as asthma or COPD, a doctor will give you an antibiotic just in case or to nip in in the bud if it is a bacteria.
Without being in the doctors office with you, or seeing you personally, it's hard for me to really know what your doctor was thinking. Yet in this answer I provided you with a couple of options.
What you'll need to do now (at least in my opinion) is call your doctor back and tell him his antibiotic didn't work. He should have other options for you.
Good luck. Rick.