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.

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.