Your 13 week old puppy is in puppy coat and probably is not going to look shiny and silky unless they get the silk coat that comes in as an adult. What I am saying is puppy coat takes a while to grow out. As long as you keep it clean and brushed every day to distribute the natural oil in their coat, that is what is going to happen. Also, feeding a good quality food is important to coat care. Nutrition is a big part of a good/healthy looking coat. Not just what you use to bathe them in.
All that being said, I recommend Espree products. You can look at their website.
Don't over bathe or condition your puppy or you'll just dry out his/her coat.