Properties

Label 4-288e2-1.1-c5e2-0-1
Degree 44
Conductor 8294482944
Sign 11
Analytic cond. 2133.562133.56
Root an. cond. 6.796366.79636
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  − 36·5-s − 120·7-s + 200·11-s + 284·13-s − 2.67e3·17-s + 72·19-s − 3.84e3·23-s + 2.65e3·25-s − 1.02e4·29-s + 1.04e4·31-s + 4.32e3·35-s + 1.31e4·37-s − 4.16e3·41-s − 5.83e3·43-s − 1.52e3·47-s − 1.48e4·49-s − 9.01e3·53-s − 7.20e3·55-s + 5.50e4·59-s − 6.34e4·61-s − 1.02e4·65-s + 3.67e4·67-s − 3.76e4·71-s − 3.78e4·73-s − 2.40e4·77-s + 1.44e5·79-s + 1.09e5·83-s + ⋯
L(s)  = 1  − 0.643·5-s − 0.925·7-s + 0.498·11-s + 0.466·13-s − 2.24·17-s + 0.0457·19-s − 1.51·23-s + 0.850·25-s − 2.25·29-s + 1.96·31-s + 0.596·35-s + 1.57·37-s − 0.386·41-s − 0.481·43-s − 0.100·47-s − 0.885·49-s − 0.440·53-s − 0.320·55-s + 2.06·59-s − 2.18·61-s − 0.300·65-s + 1.00·67-s − 0.886·71-s − 0.830·73-s − 0.461·77-s + 2.61·79-s + 1.74·83-s + ⋯

Functional equation

Λ(s)=(82944s/2ΓC(s)2L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}
Λ(s)=(82944s/2ΓC(s+5/2)2L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 82944 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 8294482944    =    210342^{10} \cdot 3^{4}
Sign: 11
Analytic conductor: 2133.562133.56
Root analytic conductor: 6.796366.79636
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 82944, ( :5/2,5/2), 1)(4,\ 82944,\ (\ :5/2, 5/2),\ 1)

Particular Values

L(3)L(3) \approx 0.25906485980.2590648598
L(12)L(\frac12) \approx 0.25906485980.2590648598
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)
bad2 1 1
3 1 1
good5D4D_{4} 1+36T1362T2+36p5T3+p10T4 1 + 36 T - 1362 T^{2} + 36 p^{5} T^{3} + p^{10} T^{4}
7D4D_{4} 1+120T+29278T2+120p5T3+p10T4 1 + 120 T + 29278 T^{2} + 120 p^{5} T^{3} + p^{10} T^{4}
11D4D_{4} 1200T+46406T2200p5T3+p10T4 1 - 200 T + 46406 T^{2} - 200 p^{5} T^{3} + p^{10} T^{4}
13D4D_{4} 1284T+477054T2284p5T3+p10T4 1 - 284 T + 477054 T^{2} - 284 p^{5} T^{3} + p^{10} T^{4}
17D4D_{4} 1+2676T+4344262T2+2676p5T3+p10T4 1 + 2676 T + 4344262 T^{2} + 2676 p^{5} T^{3} + p^{10} T^{4}
19D4D_{4} 172T+4159894T272p5T3+p10T4 1 - 72 T + 4159894 T^{2} - 72 p^{5} T^{3} + p^{10} T^{4}
23D4D_{4} 1+3840T+9416686T2+3840p5T3+p10T4 1 + 3840 T + 9416686 T^{2} + 3840 p^{5} T^{3} + p^{10} T^{4}
29D4D_{4} 1+10212T+64228638T2+10212p5T3+p10T4 1 + 10212 T + 64228638 T^{2} + 10212 p^{5} T^{3} + p^{10} T^{4}
31D4D_{4} 110488T+84559438T210488p5T3+p10T4 1 - 10488 T + 84559438 T^{2} - 10488 p^{5} T^{3} + p^{10} T^{4}
37D4D_{4} 113148T+125908974T213148p5T3+p10T4 1 - 13148 T + 125908974 T^{2} - 13148 p^{5} T^{3} + p^{10} T^{4}
41D4D_{4} 1+4164T+216207126T2+4164p5T3+p10T4 1 + 4164 T + 216207126 T^{2} + 4164 p^{5} T^{3} + p^{10} T^{4}
43D4D_{4} 1+5832T+168401542T2+5832p5T3+p10T4 1 + 5832 T + 168401542 T^{2} + 5832 p^{5} T^{3} + p^{10} T^{4}
47D4D_{4} 1+1520T+148144670T2+1520p5T3+p10T4 1 + 1520 T + 148144670 T^{2} + 1520 p^{5} T^{3} + p^{10} T^{4}
53D4D_{4} 1+9012T+816689646T2+9012p5T3+p10T4 1 + 9012 T + 816689646 T^{2} + 9012 p^{5} T^{3} + p^{10} T^{4}
59D4D_{4} 155096T+1818478886T255096p5T3+p10T4 1 - 55096 T + 1818478886 T^{2} - 55096 p^{5} T^{3} + p^{10} T^{4}
61D4D_{4} 1+63444T+2677193342T2+63444p5T3+p10T4 1 + 63444 T + 2677193342 T^{2} + 63444 p^{5} T^{3} + p^{10} T^{4}
67D4D_{4} 136792T+1148752246T236792p5T3+p10T4 1 - 36792 T + 1148752246 T^{2} - 36792 p^{5} T^{3} + p^{10} T^{4}
71D4D_{4} 1+37664T+2440628942T2+37664p5T3+p10T4 1 + 37664 T + 2440628942 T^{2} + 37664 p^{5} T^{3} + p^{10} T^{4}
73D4D_{4} 1+37836T+2085902966T2+37836p5T3+p10T4 1 + 37836 T + 2085902966 T^{2} + 37836 p^{5} T^{3} + p^{10} T^{4}
79D4D_{4} 1144888T+10711615534T2144888p5T3+p10T4 1 - 144888 T + 10711615534 T^{2} - 144888 p^{5} T^{3} + p^{10} T^{4}
83D4D_{4} 1109272T+10472055958T2109272p5T3+p10T4 1 - 109272 T + 10472055958 T^{2} - 109272 p^{5} T^{3} + p^{10} T^{4}
89D4D_{4} 132556T+8836559958T232556p5T3+p10T4 1 - 32556 T + 8836559958 T^{2} - 32556 p^{5} T^{3} + p^{10} T^{4}
97D4D_{4} 169092T+18339537030T269092p5T3+p10T4 1 - 69092 T + 18339537030 T^{2} - 69092 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

−11.10349898750354388844759098873, −10.99592506406160479889289097081, −10.15761506248492035931915018225, −9.813597982188160246565459516624, −9.207721128276852356687927329682, −8.951303769847119867169886672344, −8.304242936937685415001827320874, −7.88489096029691078771304770369, −7.32347867296518986850370489202, −6.62992318519195227892801964667, −6.25151844223162851296684755208, −6.10173524444721992750269603546, −4.91926282841069563636886183455, −4.57694241511532075752866878131, −3.71737995047683625391745563519, −3.67160078121328846309029964919, −2.59405891947027587264839096536, −2.09299084263045959209608581651, −1.11808101953705506783798796545, −0.15018176816877938014164731810, 0.15018176816877938014164731810, 1.11808101953705506783798796545, 2.09299084263045959209608581651, 2.59405891947027587264839096536, 3.67160078121328846309029964919, 3.71737995047683625391745563519, 4.57694241511532075752866878131, 4.91926282841069563636886183455, 6.10173524444721992750269603546, 6.25151844223162851296684755208, 6.62992318519195227892801964667, 7.32347867296518986850370489202, 7.88489096029691078771304770369, 8.304242936937685415001827320874, 8.951303769847119867169886672344, 9.207721128276852356687927329682, 9.813597982188160246565459516624, 10.15761506248492035931915018225, 10.99592506406160479889289097081, 11.10349898750354388844759098873

Graph of the ZZ-function along the critical line