As far as I know, the precise conjecture for what you are asking is:
All the elements of the Selberg class that are not Artin L-functions are:
motivic L-functions of dimension bigger than 0.
transcendental L-functions
Note that Artin L-functions are the 0-dimensional motivic L-functions.
Also, transcendental L-function is an umbrella term for automorphic L-functions that are not motivic.
