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Properties

Label 4-87e4-1.1-c1e2-0-1
Degree 44
Conductor 5728976157289761
Sign 11
Analytic cond. 3652.843652.84
Root an. cond. 7.774237.77423
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-s − 2·4-s − 5-s − 3·8-s − 10-s + 5·11-s + 4·13-s + 16-s + 11·17-s − 3·19-s + 2·20-s + 5·22-s − 2·23-s + 2·25-s + 4·26-s − 9·31-s + 2·32-s + 11·34-s − 4·37-s − 3·38-s + 3·40-s − 41-s − 10·43-s − 10·44-s − 2·46-s + 14·47-s − 9·49-s + ⋯
L(s)  = 1  + 0.707·2-s − 4-s − 0.447·5-s − 1.06·8-s − 0.316·10-s + 1.50·11-s + 1.10·13-s + 1/4·16-s + 2.66·17-s − 0.688·19-s + 0.447·20-s + 1.06·22-s − 0.417·23-s + 2/5·25-s + 0.784·26-s − 1.61·31-s + 0.353·32-s + 1.88·34-s − 0.657·37-s − 0.486·38-s + 0.474·40-s − 0.156·41-s − 1.52·43-s − 1.50·44-s − 0.294·46-s + 2.04·47-s − 9/7·49-s + ⋯

Functional equation

Λ(s)=(57289761s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(57289761s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 57289761 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 5728976157289761    =    34⋅2943^{4} \cdot 29^{4}
Sign: 11
Analytic conductor: 3652.843652.84
Root analytic conductor: 7.774237.77423
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 57289761, ( :1/2,1/2), 1)(4,\ 57289761,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6047853061.604785306
L(12)L(\frac12) ≈\approx 1.6047853061.604785306
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)
bad3 1 1
29 1 1
good2D4D_{4} 1−T+3T2−pT3+p2T4 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4}
5D4D_{4} 1+T−T2+pT3+p2T4 1 + T - T^{2} + p T^{3} + p^{2} T^{4}
7C22C_2^2 1+9T2+p2T4 1 + 9 T^{2} + p^{2} T^{4}
11D4D_{4} 1−5T+27T2−5pT3+p2T4 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4}
13D4D_{4} 1−4T+25T2−4pT3+p2T4 1 - 4 T + 25 T^{2} - 4 p T^{3} + p^{2} T^{4}
17D4D_{4} 1−11T+63T2−11pT3+p2T4 1 - 11 T + 63 T^{2} - 11 p T^{3} + p^{2} T^{4}
19D4D_{4} 1+3T+29T2+3pT3+p2T4 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4}
23D4D_{4} 1+2T+42T2+2pT3+p2T4 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4}
31D4D_{4} 1+9T+51T2+9pT3+p2T4 1 + 9 T + 51 T^{2} + 9 p T^{3} + p^{2} T^{4}
37D4D_{4} 1+4T+33T2+4pT3+p2T4 1 + 4 T + 33 T^{2} + 4 p T^{3} + p^{2} T^{4}
41D4D_{4} 1+T+71T2+pT3+p2T4 1 + T + 71 T^{2} + p T^{3} + p^{2} T^{4}
43D4D_{4} 1+10T+106T2+10pT3+p2T4 1 + 10 T + 106 T^{2} + 10 p T^{3} + p^{2} T^{4}
47C2C_2 (1−7T+pT2)2 ( 1 - 7 T + p T^{2} )^{2}
53C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
59D4D_{4} 1+T+87T2+pT3+p2T4 1 + T + 87 T^{2} + p T^{3} + p^{2} T^{4}
61D4D_{4} 1−T+121T2−pT3+p2T4 1 - T + 121 T^{2} - p T^{3} + p^{2} T^{4}
67D4D_{4} 1+12T+150T2+12pT3+p2T4 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4}
71D4D_{4} 1+12T+158T2+12pT3+p2T4 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4}
73D4D_{4} 1+14T+150T2+14pT3+p2T4 1 + 14 T + 150 T^{2} + 14 p T^{3} + p^{2} T^{4}
79D4D_{4} 1+T+127T2+pT3+p2T4 1 + T + 127 T^{2} + p T^{3} + p^{2} T^{4}
83D4D_{4} 1−2T+87T2−2pT3+p2T4 1 - 2 T + 87 T^{2} - 2 p T^{3} + p^{2} T^{4}
89D4D_{4} 1−4T+137T2−4pT3+p2T4 1 - 4 T + 137 T^{2} - 4 p T^{3} + p^{2} T^{4}
97D4D_{4} 1+13T+135T2+13pT3+p2T4 1 + 13 T + 135 T^{2} + 13 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

−7.88459943534720628157692914467, −7.83537674822379891727120988647, −7.44017077797015206223541996258, −6.84037561779015388678618355210, −6.72017767568210285323956542016, −6.22673600252947180946836472461, −5.78410466072182894272959827141, −5.53140973067693156846699179620, −5.35783377548697195677269349012, −4.88793211929119153623588473287, −4.26173422144343011549242531135, −4.13267674993182048909767688163, −3.70633238115869278558748447001, −3.68208429220805701304412300669, −3.00317137124081724324657105516, −2.90569629734524821377489212328, −1.78482537912958157657611169535, −1.43571576194055332557475879467, −1.17282855779647595164503915840, −0.29756994223083770618652804036, 0.29756994223083770618652804036, 1.17282855779647595164503915840, 1.43571576194055332557475879467, 1.78482537912958157657611169535, 2.90569629734524821377489212328, 3.00317137124081724324657105516, 3.68208429220805701304412300669, 3.70633238115869278558748447001, 4.13267674993182048909767688163, 4.26173422144343011549242531135, 4.88793211929119153623588473287, 5.35783377548697195677269349012, 5.53140973067693156846699179620, 5.78410466072182894272959827141, 6.22673600252947180946836472461, 6.72017767568210285323956542016, 6.84037561779015388678618355210, 7.44017077797015206223541996258, 7.83537674822379891727120988647, 7.88459943534720628157692914467

Graph of the ZZ-function along the critical line