Does this map send a member of the Selberg class to another one ?

Question

Let $ F $ be an element of the Selberg class of degree $ d>1 $ and let $ p $ a prime dividing $ d $ . Does the map $ \phi_{p} : a_{n}(F)\mapsto a_{n}(F)^{1/p} $ where $ F(s)=\sum_{n>0}\frac{a_{n}(F)}{n^s} $ for $ \Re(s)>1 $ defines another element $ \Phi_{p}(F) :s\mapsto\sum_{n}\frac{\phi_{p}(a_{n}(F))}{n^s} $ for $ \Re(s)>1 $ of the Selberg class ? If yes what is its degree ?