Throughout this question, I call 'L-function' any automorphic L-function belonging to the Selberg class.


Suppose $ (F\_i)\_{(i>0)} $ is a sequence of L-functions with $ F\_i $ of degree $ p\_i $ for all $ i $ and let $ a\_{p\_{i}} $ be the $ p\_i$ -th coefficient of the L-series defining $ F\_i $ .


Does the sequence $ a\_{p\_i} $ define an L-function ? 



Does this particular L-series built from L-functions of prime degree define an L-function?
Surely not. 


First, you can't just patch things together at random to get an $L$-function.


Second, your "construction" doesn't limit the growth of the $a\_{p\_i}$'s, so they need not satisfy Ramanujan. 


