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.