Properties

Label 4-882e2-1.1-c5e2-0-15
Degree 44
Conductor 777924777924
Sign 11
Analytic cond. 20010.520010.5
Root an. cond. 11.893611.8936
Motivic weight 55
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  + 8·2-s + 48·4-s − 108·5-s + 256·8-s − 864·10-s − 124·11-s + 720·13-s + 1.28e3·16-s + 1.26e3·17-s + 360·19-s − 5.18e3·20-s − 992·22-s − 6.52e3·23-s + 4.94e3·25-s + 5.76e3·26-s − 7.08e3·29-s + 5.90e3·31-s + 6.14e3·32-s + 1.00e4·34-s − 6.04e3·37-s + 2.88e3·38-s − 2.76e4·40-s + 1.73e4·41-s − 608·43-s − 5.95e3·44-s − 5.21e4·46-s + 3.04e4·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 1.93·5-s + 1.41·8-s − 2.73·10-s − 0.308·11-s + 1.18·13-s + 5/4·16-s + 1.05·17-s + 0.228·19-s − 2.89·20-s − 0.436·22-s − 2.57·23-s + 1.58·25-s + 1.67·26-s − 1.56·29-s + 1.10·31-s + 1.06·32-s + 1.49·34-s − 0.725·37-s + 0.323·38-s − 2.73·40-s + 1.61·41-s − 0.0501·43-s − 0.463·44-s − 3.63·46-s + 2.01·47-s + ⋯

Functional equation

Λ(s)=(777924s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(777924s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 777924777924    =    2234742^{2} \cdot 3^{4} \cdot 7^{4}
Sign: 11
Analytic conductor: 20010.520010.5
Root analytic conductor: 11.893611.8936
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 22
Selberg data: (4, 777924, ( :5/2,5/2), 1)(4,\ 777924,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) == 00
L(12)L(\frac12) == 00
L(72)L(\frac{7}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C1C_1 (1p2T)2 ( 1 - p^{2} T )^{2}
3 1 1
7 1 1
good5D4D_{4} 1+108T+6716T2+108p5T3+p10T4 1 + 108 T + 6716 T^{2} + 108 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1+124T+294194T2+124p5T3+p10T4 1 + 124 T + 294194 T^{2} + 124 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1720T+673736T2720p5T3+p10T4 1 - 720 T + 673736 T^{2} - 720 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 11260T+2796692T21260p5T3+p10T4 1 - 1260 T + 2796692 T^{2} - 1260 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 1360T+4777230T2360p5T3+p10T4 1 - 360 T + 4777230 T^{2} - 360 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+6524T+1009894pT2+6524p5T3+p10T4 1 + 6524 T + 1009894 p T^{2} + 6524 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+7088T+48216146T2+7088p5T3+p10T4 1 + 7088 T + 48216146 T^{2} + 7088 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 15904T+58828406T25904p5T3+p10T4 1 - 5904 T + 58828406 T^{2} - 5904 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 1+6040T+111103002T2+6040p5T3+p10T4 1 + 6040 T + 111103002 T^{2} + 6040 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 117388T+216792980T217388p5T3+p10T4 1 - 17388 T + 216792980 T^{2} - 17388 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+608T+164053110T2+608p5T3+p10T4 1 + 608 T + 164053110 T^{2} + 608 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1648pT+629420198T2648p6T3+p10T4 1 - 648 p T + 629420198 T^{2} - 648 p^{6} T^{3} + p^{10} T^{4}
53D4D_{4} 1+3964T+594431822T2+3964p5T3+p10T4 1 + 3964 T + 594431822 T^{2} + 3964 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 1+40752T+1841031182T2+40752p5T3+p10T4 1 + 40752 T + 1841031182 T^{2} + 40752 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+1368T+1545466296T2+1368p5T3+p10T4 1 + 1368 T + 1545466296 T^{2} + 1368 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 1+16224T+813179750T2+16224p5T3+p10T4 1 + 16224 T + 813179750 T^{2} + 16224 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 13204T212201462T23204p5T3+p10T4 1 - 3204 T - 212201462 T^{2} - 3204 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 123976T+68704368T223976p5T3+p10T4 1 - 23976 T + 68704368 T^{2} - 23976 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1+1040pT+7785668670T2+1040p6T3+p10T4 1 + 1040 p T + 7785668670 T^{2} + 1040 p^{6} T^{3} + p^{10} T^{4}
83D4D_{4} 1+173736T+14732805782T2+173736p5T3+p10T4 1 + 173736 T + 14732805782 T^{2} + 173736 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 1+200556T+21158382260T2+200556p5T3+p10T4 1 + 200556 T + 21158382260 T^{2} + 200556 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 1251928T+32348722272T2251928p5T3+p10T4 1 - 251928 T + 32348722272 T^{2} - 251928 p^{5} T^{3} + p^{10} T^{4}
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   L(s)=p j=14(1αj,pps)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

−9.047409172072625776713966181973, −8.590615522744635089646514918710, −8.016107556012015818949064595035, −7.88277245126493696555168659416, −7.38961933418289585681318955410, −7.28099661275107127614004906919, −6.31815670830155299166156541536, −6.14966986664153974324857214883, −5.54716310709169844178730456079, −5.36774037980630252284283547740, −4.37672752137482228456574187899, −4.20775747082601207721577958367, −3.74490234916446448722665794607, −3.68192868222775962197203958632, −2.81355463206960669057603142382, −2.51705136803645793995081085002, −1.47499430740729147525165745607, −1.19102157271984553626278081789, 0, 0, 1.19102157271984553626278081789, 1.47499430740729147525165745607, 2.51705136803645793995081085002, 2.81355463206960669057603142382, 3.68192868222775962197203958632, 3.74490234916446448722665794607, 4.20775747082601207721577958367, 4.37672752137482228456574187899, 5.36774037980630252284283547740, 5.54716310709169844178730456079, 6.14966986664153974324857214883, 6.31815670830155299166156541536, 7.28099661275107127614004906919, 7.38961933418289585681318955410, 7.88277245126493696555168659416, 8.016107556012015818949064595035, 8.590615522744635089646514918710, 9.047409172072625776713966181973

Graph of the ZZ-function along the critical line