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Properties

Label 4-14e4-1.1-c13e2-0-1
Degree 44
Conductor 3841638416
Sign 11
Analytic cond. 44172.544172.5
Root an. cond. 14.497314.4973
Motivic weight 1313
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  − 468·3-s − 5.62e4·5-s + 1.59e6·9-s + 6.39e6·11-s + 3.03e7·13-s + 2.63e7·15-s − 4.31e7·17-s + 3.65e8·19-s + 5.72e7·23-s + 1.22e9·25-s − 2.13e9·27-s − 9.28e7·29-s + 5.68e9·31-s − 2.99e9·33-s + 1.88e9·37-s − 1.42e10·39-s − 1.46e10·41-s − 5.37e10·43-s − 8.96e10·45-s − 1.01e11·47-s + 2.01e10·51-s − 2.78e11·53-s − 3.59e11·55-s − 1.70e11·57-s − 5.95e10·59-s + 2.74e10·61-s − 1.70e12·65-s + ⋯
L(s)  = 1  − 0.370·3-s − 1.60·5-s + 9-s + 1.08·11-s + 1.74·13-s + 0.596·15-s − 0.433·17-s + 1.78·19-s + 0.0806·23-s + 25-s − 1.06·27-s − 0.0289·29-s + 1.14·31-s − 0.403·33-s + 0.120·37-s − 0.647·39-s − 0.482·41-s − 1.29·43-s − 1.60·45-s − 1.37·47-s + 0.160·51-s − 1.72·53-s − 1.75·55-s − 0.659·57-s − 0.183·59-s + 0.0683·61-s − 2.81·65-s + ⋯

Functional equation

Λ(s)=(38416s/2 ΓC(s)2 L(s)=( Λ(14−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}
Λ(s)=(38416s/2 ΓC(s+13/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+13/2)^{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: 44172.544172.5
Root analytic conductor: 14.497314.4973
Motivic weight: 1313
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 38416, ( :13/2,13/2), 1)(4,\ 38416,\ (\ :13/2, 13/2),\ 1)

Particular Values

L(7)L(7) ≈\approx 1.9769720701.976972070
L(12)L(\frac12) ≈\approx 1.9769720701.976972070
L(152)L(\frac{15}{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+52p2T−16979p4T2+52p15T3+p26T4 1 + 52 p^{2} T - 16979 p^{4} T^{2} + 52 p^{15} T^{3} + p^{26} T^{4}
5C22C_2^2 1+56214T+1939310671T2+56214p13T3+p26T4 1 + 56214 T + 1939310671 T^{2} + 56214 p^{13} T^{3} + p^{26} T^{4}
11C22C_2^2 1−581580pT+52923625789p2T2−581580p14T3+p26T4 1 - 581580 p T + 52923625789 p^{2} T^{2} - 581580 p^{14} T^{3} + p^{26} T^{4}
13C2C_2 (1−15199742T+p13T2)2 ( 1 - 15199742 T + p^{13} T^{2} )^{2}
17C22C_2^2 1+43114194T−8045744308636301T2+43114194p13T3+p26T4 1 + 43114194 T - 8045744308636301 T^{2} + 43114194 p^{13} T^{3} + p^{26} T^{4}
19C22C_2^2 1−365115484T+91256333194297197T2−365115484p13T3+p26T4 1 - 365115484 T + 91256333194297197 T^{2} - 365115484 p^{13} T^{3} + p^{26} T^{4}
23C22C_2^2 1−57226824T−500761452551340407T2−57226824p13T3+p26T4 1 - 57226824 T - 500761452551340407 T^{2} - 57226824 p^{13} T^{3} + p^{26} T^{4}
29C2C_2 (1+46418994T+p13T2)2 ( 1 + 46418994 T + p^{13} T^{2} )^{2}
31C22C_2^2 1−5682185824T+7869689441021516385T2−5682185824p13T3+p26T4 1 - 5682185824 T + 7869689441021516385 T^{2} - 5682185824 p^{13} T^{3} + p^{26} T^{4}
37C22C_2^2 1−1887185098T− 1 - 1887185098 T - 24 ⁣⋯ ⁣9324\!\cdots\!93T2−1887185098p13T3+p26T4 T^{2} - 1887185098 p^{13} T^{3} + p^{26} T^{4}
41C2C_2 (1+7336802934T+p13T2)2 ( 1 + 7336802934 T + p^{13} T^{2} )^{2}
43C2C_2 (1+26886674980T+p13T2)2 ( 1 + 26886674980 T + p^{13} T^{2} )^{2}
47C22C_2^2 1+101839834224T+ 1 + 101839834224 T + 49 ⁣⋯ ⁣4949\!\cdots\!49T2+101839834224p13T3+p26T4 T^{2} + 101839834224 p^{13} T^{3} + p^{26} T^{4}
53C22C_2^2 1+278731884294T+ 1 + 278731884294 T + 51 ⁣⋯ ⁣6351\!\cdots\!63T2+278731884294p13T3+p26T4 T^{2} + 278731884294 p^{13} T^{3} + p^{26} T^{4}
59C22C_2^2 1+59573945772T− 1 + 59573945772 T - 10 ⁣⋯ ⁣9510\!\cdots\!95T2+59573945772p13T3+p26T4 T^{2} + 59573945772 p^{13} T^{3} + p^{26} T^{4}
61C22C_2^2 1−27484470418T− 1 - 27484470418 T - 16 ⁣⋯ ⁣5716\!\cdots\!57T2−27484470418p13T3+p26T4 T^{2} - 27484470418 p^{13} T^{3} + p^{26} T^{4}
67C22C_2^2 1+784410054932T+ 1 + 784410054932 T + 67 ⁣⋯ ⁣3767\!\cdots\!37T2+784410054932p13T3+p26T4 T^{2} + 784410054932 p^{13} T^{3} + p^{26} T^{4}
71C2C_2 (1+360365227992T+p13T2)2 ( 1 + 360365227992 T + p^{13} T^{2} )^{2}
73C22C_2^2 1−1592635413718T+ 1 - 1592635413718 T + 86 ⁣⋯ ⁣9186\!\cdots\!91T2−1592635413718p13T3+p26T4 T^{2} - 1592635413718 p^{13} T^{3} + p^{26} T^{4}
79C22C_2^2 1−23161184752T− 1 - 23161184752 T - 46 ⁣⋯ ⁣3546\!\cdots\!35T2−23161184752p13T3+p26T4 T^{2} - 23161184752 p^{13} T^{3} + p^{26} T^{4}
83C2C_2 (1−2050158110436T+p13T2)2 ( 1 - 2050158110436 T + p^{13} T^{2} )^{2}
89C22C_2^2 1−3485391237126T− 1 - 3485391237126 T - 98 ⁣⋯ ⁣9398\!\cdots\!93T2−3485391237126p13T3+p26T4 T^{2} - 3485391237126 p^{13} T^{3} + p^{26} T^{4}
97C2C_2 (1−6706667416802T+p13T2)2 ( 1 - 6706667416802 T + p^{13} 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

−10.39175975959297282209815035651, −10.05772979942860355645338792731, −9.260078736914698214567355507465, −9.100742968732378391027613994037, −8.321611803711677235152967810275, −7.77922641591313266127960166255, −7.72368874479589343267241630478, −6.81847520713792590062817560035, −6.53163629025362579850982607400, −6.13415146708623202929821027791, −5.21417823022646221929437995595, −4.77416998538238980650320505853, −4.19838810776529131789834813735, −3.65620849639025240218292073302, −3.52214098308492647886991825373, −2.87717758063919987927101152932, −1.60527647060925818743311800377, −1.47284035229354196644291753345, −0.887892527940874551558263052907, −0.32237145047148958660329505608, 0.32237145047148958660329505608, 0.887892527940874551558263052907, 1.47284035229354196644291753345, 1.60527647060925818743311800377, 2.87717758063919987927101152932, 3.52214098308492647886991825373, 3.65620849639025240218292073302, 4.19838810776529131789834813735, 4.77416998538238980650320505853, 5.21417823022646221929437995595, 6.13415146708623202929821027791, 6.53163629025362579850982607400, 6.81847520713792590062817560035, 7.72368874479589343267241630478, 7.77922641591313266127960166255, 8.321611803711677235152967810275, 9.100742968732378391027613994037, 9.260078736914698214567355507465, 10.05772979942860355645338792731, 10.39175975959297282209815035651

Graph of the ZZ-function along the critical line