In its simplest form a function f is Modular of weight k if it satisfies the following functional equation:
f(Tz) = (cz+d)^kf(z),
where T is a transformation belonging to a subgroup of SL(2,Z), such that Tz = az+b/(cz+d).
It can be shown that the vector space of analytic modular forms of weight k is of finite rank.
There are two basic analytic modular forms:
1. neta-function (also called Ramanujan Modular Form)
2. Eisenstein Series
With every analytic modular form, one can associate a zeta function, or L-form