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Properties

Label 4-7920e2-1.1-c1e2-0-7
Degree 44
Conductor 6272640062726400
Sign 11
Analytic cond. 3999.483999.48
Root an. cond. 7.952457.95245
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 4·7-s − 2·11-s + 8·19-s + 3·25-s − 4·29-s + 4·31-s + 8·35-s + 4·37-s − 4·41-s + 4·43-s + 8·47-s − 4·53-s − 4·55-s + 12·59-s + 12·61-s + 4·71-s − 8·77-s + 16·79-s + 12·83-s − 4·89-s + 16·95-s + 12·97-s − 20·101-s + 12·107-s − 4·109-s − 4·113-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s − 0.603·11-s + 1.83·19-s + 3/5·25-s − 0.742·29-s + 0.718·31-s + 1.35·35-s + 0.657·37-s − 0.624·41-s + 0.609·43-s + 1.16·47-s − 0.549·53-s − 0.539·55-s + 1.56·59-s + 1.53·61-s + 0.474·71-s − 0.911·77-s + 1.80·79-s + 1.31·83-s − 0.423·89-s + 1.64·95-s + 1.21·97-s − 1.99·101-s + 1.16·107-s − 0.383·109-s − 0.376·113-s + ⋯

Functional equation

Λ(s)=(62726400s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(62726400s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 62726400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 6272640062726400    =    28⋅34⋅52⋅1122^{8} \cdot 3^{4} \cdot 5^{2} \cdot 11^{2}
Sign: 11
Analytic conductor: 3999.483999.48
Root analytic conductor: 7.952457.95245
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 62726400, ( :1/2,1/2), 1)(4,\ 62726400,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 6.3921997396.392199739
L(12)L(\frac12) ≈\approx 6.3921997396.392199739
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2 1 1
3 1 1
5C1C_1 (1−T)2 ( 1 - T )^{2}
11C1C_1 (1+T)2 ( 1 + T )^{2}
good7D4D_{4} 1−4T+16T2−4pT3+p2T4 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4}
13C22C_2^2 1+24T2+p2T4 1 + 24 T^{2} + p^{2} T^{4}
17C22C_2^2 1+16T2+p2T4 1 + 16 T^{2} + p^{2} T^{4}
19D4D_{4} 1−8T+46T2−8pT3+p2T4 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4}
23C22C_2^2 1+38T2+p2T4 1 + 38 T^{2} + p^{2} T^{4}
29D4D_{4} 1+4T+30T2+4pT3+p2T4 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4}
31D4D_{4} 1−4T+58T2−4pT3+p2T4 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4}
37D4D_{4} 1−4T+46T2−4pT3+p2T4 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+4T+54T2+4pT3+p2T4 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4}
43D4D_{4} 1−4T+40T2−4pT3+p2T4 1 - 4 T + 40 T^{2} - 4 p T^{3} + p^{2} T^{4}
47D4D_{4} 1−8T+38T2−8pT3+p2T4 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4}
53D4D_{4} 1+4T+38T2+4pT3+p2T4 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4}
59D4D_{4} 1−12T+146T2−12pT3+p2T4 1 - 12 T + 146 T^{2} - 12 p T^{3} + p^{2} T^{4}
61D4D_{4} 1−12T+150T2−12pT3+p2T4 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4}
67C22C_2^2 1+62T2+p2T4 1 + 62 T^{2} + p^{2} T^{4}
71D4D_{4} 1−4T+74T2−4pT3+p2T4 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4}
73C22C_2^2 1+96T2+p2T4 1 + 96 T^{2} + p^{2} T^{4}
79D4D_{4} 1−16T+150T2−16pT3+p2T4 1 - 16 T + 150 T^{2} - 16 p T^{3} + p^{2} T^{4}
83D4D_{4} 1−12T+184T2−12pT3+p2T4 1 - 12 T + 184 T^{2} - 12 p T^{3} + p^{2} T^{4}
89C2C_2 (1+2T+pT2)2 ( 1 + 2 T + p T^{2} )^{2}
97D4D_{4} 1−12T+222T2−12pT3+p2T4 1 - 12 T + 222 T^{2} - 12 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−7.86742359342845289971225402962, −7.84788073694050585644894833015, −7.35168742222963915324465063029, −7.08494715646822316997389985779, −6.50654914424554425856181397563, −6.43249903475446675747119669406, −5.70387726442982119002732516346, −5.52082698066106743555588560117, −5.18362500483067615029293687711, −5.12527526823527625714200989903, −4.45775520727538247925520386588, −4.32246301787294433728658447599, −3.53046985465691196970212048177, −3.45290856677079954675668782919, −2.62095387748962369119489522305, −2.56220393342996253562323725020, −1.86125905996384875704005040948, −1.70853975197037744690635787908, −0.909602297106672650127731883100, −0.73290838982016040638422844127, 0.73290838982016040638422844127, 0.909602297106672650127731883100, 1.70853975197037744690635787908, 1.86125905996384875704005040948, 2.56220393342996253562323725020, 2.62095387748962369119489522305, 3.45290856677079954675668782919, 3.53046985465691196970212048177, 4.32246301787294433728658447599, 4.45775520727538247925520386588, 5.12527526823527625714200989903, 5.18362500483067615029293687711, 5.52082698066106743555588560117, 5.70387726442982119002732516346, 6.43249903475446675747119669406, 6.50654914424554425856181397563, 7.08494715646822316997389985779, 7.35168742222963915324465063029, 7.84788073694050585644894833015, 7.86742359342845289971225402962

Graph of the ZZ-function along the critical line