with a character 
modulo q.
All known examples of L-functions of degree 2 are associated to newforms
on the group with a character modulo q.
There are two basic types: holomorphic and non-holomorphic. Both are characterized by their eigenvalue set.
The holomorphic newforms have eigenvalue set
of the shape 
where k is a positive integer.
The association is via
where
is a holomorphic cusp form for a congruence subgroup of 
.
We work with L(s) because the coefficients 
are algebraic integers.
The
non-holomorphic newforms have eigenvalue set 
where u is 0 or 1/2 and where 
is presumed transcendental.