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Properties

Label 4-1200e2-1.1-c3e2-0-28
Degree 44
Conductor 14400001440000
Sign 11
Analytic cond. 5012.965012.96
Root an. cond. 8.414408.41440
Motivic weight 33
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 22

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 6·3-s + 2·7-s + 27·9-s − 74·11-s + 98·13-s − 78·17-s + 80·19-s − 12·21-s − 40·23-s − 108·27-s + 50·29-s + 12·31-s + 444·33-s − 34·37-s − 588·39-s + 344·41-s − 216·43-s − 876·47-s + 478·49-s + 468·51-s − 634·53-s − 480·57-s − 666·59-s + 244·61-s + 54·63-s + 980·67-s + 240·69-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.107·7-s + 9-s − 2.02·11-s + 2.09·13-s − 1.11·17-s + 0.965·19-s − 0.124·21-s − 0.362·23-s − 0.769·27-s + 0.320·29-s + 0.0695·31-s + 2.34·33-s − 0.151·37-s − 2.41·39-s + 1.31·41-s − 0.766·43-s − 2.71·47-s + 1.39·49-s + 1.28·51-s − 1.64·53-s − 1.11·57-s − 1.46·59-s + 0.512·61-s + 0.107·63-s + 1.78·67-s + 0.418·69-s + ⋯

Functional equation

Λ(s)=(1440000s/2 ΓC(s)2 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1440000s/2 ΓC(s+3/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 14400001440000    =    28⋅32⋅542^{8} \cdot 3^{2} \cdot 5^{4}
Sign: 11
Analytic conductor: 5012.965012.96
Root analytic conductor: 8.414408.41440
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 1440000, ( :3/2,3/2), 1)(4,\ 1440000,\ (\ :3/2, 3/2),\ 1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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
3C1C_1 (1+pT)2 ( 1 + p T )^{2}
5 1 1
good7D4D_{4} 1−2T−474T2−2p3T3+p6T4 1 - 2 T - 474 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4}
11D4D_{4} 1+74T+3902T2+74p3T3+p6T4 1 + 74 T + 3902 T^{2} + 74 p^{3} T^{3} + p^{6} T^{4}
13D4D_{4} 1−98T+6666T2−98p3T3+p6T4 1 - 98 T + 6666 T^{2} - 98 p^{3} T^{3} + p^{6} T^{4}
17D4D_{4} 1+78T+8122T2+78p3T3+p6T4 1 + 78 T + 8122 T^{2} + 78 p^{3} T^{3} + p^{6} T^{4}
19D4D_{4} 1−80T+7062T2−80p3T3+p6T4 1 - 80 T + 7062 T^{2} - 80 p^{3} T^{3} + p^{6} T^{4}
23D4D_{4} 1+40T+16478T2+40p3T3+p6T4 1 + 40 T + 16478 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4}
29D4D_{4} 1−50T+43082T2−50p3T3+p6T4 1 - 50 T + 43082 T^{2} - 50 p^{3} T^{3} + p^{6} T^{4}
31D4D_{4} 1−12T+17822T2−12p3T3+p6T4 1 - 12 T + 17822 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4}
37D4D_{4} 1+34T+20970T2+34p3T3+p6T4 1 + 34 T + 20970 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4}
41D4D_{4} 1−344T+162782T2−344p3T3+p6T4 1 - 344 T + 162782 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4}
43C2C_2 (1+108T+p3T2)2 ( 1 + 108 T + p^{3} T^{2} )^{2}
47D4D_{4} 1+876T+374206T2+876p3T3+p6T4 1 + 876 T + 374206 T^{2} + 876 p^{3} T^{3} + p^{6} T^{4}
53D4D_{4} 1+634T+398114T2+634p3T3+p6T4 1 + 634 T + 398114 T^{2} + 634 p^{3} T^{3} + p^{6} T^{4}
59D4D_{4} 1+666T+211918T2+666p3T3+p6T4 1 + 666 T + 211918 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4}
61D4D_{4} 1−4pT+336750T2−4p4T3+p6T4 1 - 4 p T + 336750 T^{2} - 4 p^{4} T^{3} + p^{6} T^{4}
67D4D_{4} 1−980T+692502T2−980p3T3+p6T4 1 - 980 T + 692502 T^{2} - 980 p^{3} T^{3} + p^{6} T^{4}
71D4D_{4} 1+308T+553262T2+308p3T3+p6T4 1 + 308 T + 553262 T^{2} + 308 p^{3} T^{3} + p^{6} T^{4}
73D4D_{4} 1+1412T+1090194T2+1412p3T3+p6T4 1 + 1412 T + 1090194 T^{2} + 1412 p^{3} T^{3} + p^{6} T^{4}
79D4D_{4} 1+1052T+1237470T2+1052p3T3+p6T4 1 + 1052 T + 1237470 T^{2} + 1052 p^{3} T^{3} + p^{6} T^{4}
83D4D_{4} 1+248T+1125926T2+248p3T3+p6T4 1 + 248 T + 1125926 T^{2} + 248 p^{3} T^{3} + p^{6} T^{4}
89D4D_{4} 1−684T+1229686T2−684p3T3+p6T4 1 - 684 T + 1229686 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4}
97D4D_{4} 1+1840T+2539650T2+1840p3T3+p6T4 1 + 1840 T + 2539650 T^{2} + 1840 p^{3} T^{3} + p^{6} 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

−9.134813727695630941301129936501, −8.727062162832495033338205550249, −8.188690276151103822548954148946, −8.095190877714211151420413806313, −7.42297558383710381328696778612, −7.16589876695208150011388504491, −6.35340026426278654169893536516, −6.35017905904169281722660168873, −5.72686912879348589735299017903, −5.49027878289581855034176344125, −4.80012424507694010567812018706, −4.69043599042521821225027955883, −3.99947599999390154890199283145, −3.40392273609469938900847431542, −2.92388713813338218541071789963, −2.27156924008469058727652243188, −1.47398797015446986921498309019, −1.10412142721995947334974773280, 0, 0, 1.10412142721995947334974773280, 1.47398797015446986921498309019, 2.27156924008469058727652243188, 2.92388713813338218541071789963, 3.40392273609469938900847431542, 3.99947599999390154890199283145, 4.69043599042521821225027955883, 4.80012424507694010567812018706, 5.49027878289581855034176344125, 5.72686912879348589735299017903, 6.35017905904169281722660168873, 6.35340026426278654169893536516, 7.16589876695208150011388504491, 7.42297558383710381328696778612, 8.095190877714211151420413806313, 8.188690276151103822548954148946, 8.727062162832495033338205550249, 9.134813727695630941301129936501

Graph of the ZZ-function along the critical line