(2,2,{(k-1)/4,(k+1)/4},±(-1)^(k/2))

Holomorphic, Degree 2, Level 2

• Weight 8 , 10 , 12 , 14 , 16 , 18 , 20 , 22 , 24 , 26 , 28 , 30 , 32 , 34 , 36 , 40 , 42 , 48
• Hecke polynomials

Back to Degree 2, Back to the start

L-functions of degree 2 and level 2 correspond to newforms on . They can be expressed as linear combinations of products of the basic Eisenstein series which we now introduce. First of all,

is the Eisenstein series of weight 2. It corresponds to the Dirichlet series

which, after normalization, satisfies a functional equation of type but is not in the Selberg class. For even weights greater than 2 we can construct modular forms from the Eisenstein series on the full modular group. Thus, for k an even integer, we let

and

These are modular forms of weight k for which have multiplicative coefficients and which are eigenforms of the matrix with eigenvalue +1 or -1 according to the superscript on the E. corresponds to the Dirichlet series

which, after normalization, has functional equation of type and corresponds to the Dirichlet series

which, after normalization, has functional equation of type Thus,