|
gptkbp:instanceOf
|
modular group
|
|
gptkbp:alsoKnownAs
|
gptkb:modular_discriminant
|
|
gptkbp:appearsIn
|
Ramanujan's 1916 paper on modular forms
|
|
gptkbp:cusp_form
|
true
|
|
gptkbp:definedIn
|
Δ(z) = q ∏_{n=1}^∞ (1 - q^n)^{24}, q = e^{2πiz}
|
|
gptkbp:discoveredBy
|
gptkb:Srinivasa_Ramanujan
|
|
gptkbp:field
|
modular forms
number theory
|
|
gptkbp:firstPublished
|
1916
|
|
gptkbp:Fourier_coefficient
|
gptkb:Ramanujan_tau_function
|
|
gptkbp:Fourier_expansion
|
Δ(z) = ∑_{n=1}^∞ τ(n)q^n
|
|
gptkbp:growthForm
|
bounded by O(n^{11/2}) for τ(n)
|
|
gptkbp:Hecke_eigenform
|
true
|
|
gptkbp:holomorphic
|
true
|
|
https://www.w3.org/2000/01/rdf-schema#label
|
Ramanujan Delta function
|
|
gptkbp:L-function
|
L(Δ,s) = ∑_{n=1}^∞ τ(n)n^{-s}
|
|
gptkbp:level
|
1
|
|
gptkbp:modular_group_action
|
Δ(γz) = (cz+d)^{12}Δ(z) for γ in SL(2,ℤ)
|
|
gptkbp:multiplicative_property
|
τ(mn) = τ(m)τ(n) for coprime m, n
|
|
gptkbp:nonzero_elsewhere
|
true
|
|
gptkbp:relatedTo
|
gptkb:modular_group_SL(2,ℤ)
gptkb:Eisenstein_series
gptkb:Ramanujan_tau_function
|
|
gptkbp:satisfies_functional_equation
|
true
|
|
gptkbp:symbol
|
Δ(z)
|
|
gptkbp:vanishes_at_infinity
|
true
|
|
gptkbp:weight
|
12
|
|
gptkbp:zeroes
|
at the cusp only
|
|
gptkbp:bfsParent
|
gptkb:cusp_form
|
|
gptkbp:bfsLayer
|
6
|