↑

Properties

Label 4-7920e2-1.1-c1e2-0-9
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

Learn more

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 4.7122737544.712273754
L(12)L(\frac12) ≈\approx 4.7122737544.712273754
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}
show more
show less
   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

−7.915267519737862949108166486253, −7.84196533970187846806394745874, −7.27455231148917980663780008306, −7.22706338004649759920767944024, −6.62579518065423300431740768651, −6.39122541649336575227585235504, −5.86797783638690144080705637099, −5.57556387930043359208140976744, −4.98992372658770539455532675687, −4.93155828520612439575850495942, −4.39770839082835835666110504710, −4.33628768835155389199481604448, −3.59228961202000058700130130778, −3.47615209050170800935456851052, −2.83488744193346577323784261823, −2.63328086337728497804891513776, −1.72935820272930836841945637350, −1.64536728341114238413282952998, −0.77476658773066623187586469196, −0.75366034043447184966304522189, 0.75366034043447184966304522189, 0.77476658773066623187586469196, 1.64536728341114238413282952998, 1.72935820272930836841945637350, 2.63328086337728497804891513776, 2.83488744193346577323784261823, 3.47615209050170800935456851052, 3.59228961202000058700130130778, 4.33628768835155389199481604448, 4.39770839082835835666110504710, 4.93155828520612439575850495942, 4.98992372658770539455532675687, 5.57556387930043359208140976744, 5.86797783638690144080705637099, 6.39122541649336575227585235504, 6.62579518065423300431740768651, 7.22706338004649759920767944024, 7.27455231148917980663780008306, 7.84196533970187846806394745874, 7.915267519737862949108166486253

Graph of the ZZ-function along the critical line