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Properties

Label 4-14e4-1.1-c5e2-0-3
Degree 44
Conductor 3841638416
Sign 11
Analytic cond. 988.173988.173
Root an. cond. 5.606715.60671
Motivic weight 55
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  − 12·3-s + 54·5-s + 243·9-s − 540·11-s + 836·13-s − 648·15-s + 594·17-s + 836·19-s + 4.10e3·23-s + 3.12e3·25-s − 7.02e3·27-s − 1.18e3·29-s + 4.25e3·31-s + 6.48e3·33-s + 298·37-s − 1.00e4·39-s − 3.44e4·41-s − 2.42e4·43-s + 1.31e4·45-s − 1.29e3·47-s − 7.12e3·51-s − 1.94e4·53-s − 2.91e4·55-s − 1.00e4·57-s − 7.66e3·59-s − 3.47e4·61-s + 4.51e4·65-s + ⋯
L(s)  = 1  − 0.769·3-s + 0.965·5-s + 9-s − 1.34·11-s + 1.37·13-s − 0.743·15-s + 0.498·17-s + 0.531·19-s + 1.61·23-s + 25-s − 1.85·27-s − 0.262·29-s + 0.795·31-s + 1.03·33-s + 0.0357·37-s − 1.05·39-s − 3.20·41-s − 1.99·43-s + 0.965·45-s − 0.0855·47-s − 0.383·51-s − 0.953·53-s − 1.29·55-s − 0.408·57-s − 0.286·59-s − 1.19·61-s + 1.32·65-s + ⋯

Functional equation

Λ(s)=(38416s/2 ΓC(s)2 L(s)=( Λ(6−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(38416s/2 ΓC(s+5/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 3841638416    =    24⋅742^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 988.173988.173
Root analytic conductor: 5.606715.60671
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 38416, ( :5/2,5/2), 1)(4,\ 38416,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) ≈\approx 2.1880323262.188032326
L(12)L(\frac12) ≈\approx 2.1880323262.188032326
L(72)L(\frac{7}{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
7 1 1
good3C22C_2^2 1+4pT−11p2T2+4p6T3+p10T4 1 + 4 p T - 11 p^{2} T^{2} + 4 p^{6} T^{3} + p^{10} T^{4}
5C22C_2^2 1−54T−209T2−54p5T3+p10T4 1 - 54 T - 209 T^{2} - 54 p^{5} T^{3} + p^{10} T^{4}
11C22C_2^2 1+540T+130549T2+540p5T3+p10T4 1 + 540 T + 130549 T^{2} + 540 p^{5} T^{3} + p^{10} T^{4}
13C2C_2 (1−418T+p5T2)2 ( 1 - 418 T + p^{5} T^{2} )^{2}
17C22C_2^2 1−594T−1067021T2−594p5T3+p10T4 1 - 594 T - 1067021 T^{2} - 594 p^{5} T^{3} + p^{10} T^{4}
19C22C_2^2 1−44pT−4923p2T2−44p6T3+p10T4 1 - 44 p T - 4923 p^{2} T^{2} - 44 p^{6} T^{3} + p^{10} T^{4}
23C22C_2^2 1−4104T+10406473T2−4104p5T3+p10T4 1 - 4104 T + 10406473 T^{2} - 4104 p^{5} T^{3} + p^{10} T^{4}
29C2C_2 (1+594T+p5T2)2 ( 1 + 594 T + p^{5} T^{2} )^{2}
31C22C_2^2 1−4256T−10515615T2−4256p5T3+p10T4 1 - 4256 T - 10515615 T^{2} - 4256 p^{5} T^{3} + p^{10} T^{4}
37C22C_2^2 1−298T−69255153T2−298p5T3+p10T4 1 - 298 T - 69255153 T^{2} - 298 p^{5} T^{3} + p^{10} T^{4}
41C2C_2 (1+17226T+p5T2)2 ( 1 + 17226 T + p^{5} T^{2} )^{2}
43C2C_2 (1+12100T+p5T2)2 ( 1 + 12100 T + p^{5} T^{2} )^{2}
47C22C_2^2 1+1296T−227665391T2+1296p5T3+p10T4 1 + 1296 T - 227665391 T^{2} + 1296 p^{5} T^{3} + p^{10} T^{4}
53C22C_2^2 1+19494T−38179457T2+19494p5T3+p10T4 1 + 19494 T - 38179457 T^{2} + 19494 p^{5} T^{3} + p^{10} T^{4}
59C22C_2^2 1+7668T−656126075T2+7668p5T3+p10T4 1 + 7668 T - 656126075 T^{2} + 7668 p^{5} T^{3} + p^{10} T^{4}
61C22C_2^2 1+34738T+362132343T2+34738p5T3+p10T4 1 + 34738 T + 362132343 T^{2} + 34738 p^{5} T^{3} + p^{10} T^{4}
67C22C_2^2 1+21812T−874361763T2+21812p5T3+p10T4 1 + 21812 T - 874361763 T^{2} + 21812 p^{5} T^{3} + p^{10} T^{4}
71C2C_2 (1+46872T+p5T2)2 ( 1 + 46872 T + p^{5} T^{2} )^{2}
73C22C_2^2 1−67562T+2491552251T2−67562p5T3+p10T4 1 - 67562 T + 2491552251 T^{2} - 67562 p^{5} T^{3} + p^{10} T^{4}
79C22C_2^2 1−76912T+2838399345T2−76912p5T3+p10T4 1 - 76912 T + 2838399345 T^{2} - 76912 p^{5} T^{3} + p^{10} T^{4}
83C2C_2 (1+67716T+p5T2)2 ( 1 + 67716 T + p^{5} T^{2} )^{2}
89C22C_2^2 1−29754T−4698758933T2−29754p5T3+p10T4 1 - 29754 T - 4698758933 T^{2} - 29754 p^{5} T^{3} + p^{10} T^{4}
97C2C_2 (1−122398T+p5T2)2 ( 1 - 122398 T + p^{5} T^{2} )^{2}
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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

−11.85119107672026696646535185627, −11.26209083439098568390253517226, −10.87882798990519051768630790944, −10.26348006073767225104631589088, −10.14860774335752357848964180539, −9.503386557880415555607349487656, −8.950271390804328155913123521383, −8.279499185353390168756216445291, −7.87666714272711221776639454973, −6.92526924955320658115343795397, −6.84012529087407548993483271901, −5.93142795273266138445825588925, −5.64245290954351691803255546160, −4.92465214809842891543103173085, −4.68977070612841135500151565035, −3.24102611726558864429364232170, −3.24064691863337616706659543766, −1.79740224768445110230891669265, −1.45734951064192139503349566474, −0.49213098530591072928089558796, 0.49213098530591072928089558796, 1.45734951064192139503349566474, 1.79740224768445110230891669265, 3.24064691863337616706659543766, 3.24102611726558864429364232170, 4.68977070612841135500151565035, 4.92465214809842891543103173085, 5.64245290954351691803255546160, 5.93142795273266138445825588925, 6.84012529087407548993483271901, 6.92526924955320658115343795397, 7.87666714272711221776639454973, 8.279499185353390168756216445291, 8.950271390804328155913123521383, 9.503386557880415555607349487656, 10.14860774335752357848964180539, 10.26348006073767225104631589088, 10.87882798990519051768630790944, 11.26209083439098568390253517226, 11.85119107672026696646535185627

Graph of the ZZ-function along the critical line