Properties

Label 8-588e4-1.1-c5e4-0-3
Degree 88
Conductor 119538913536119538913536
Sign 11
Analytic cond. 7.90954×1077.90954\times 10^{7}
Root an. cond. 9.711119.71111
Motivic weight 55
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 18·3-s − 6·5-s + 81·9-s + 90·11-s − 1.53e3·13-s + 108·15-s + 1.92e3·17-s + 2.24e3·19-s − 6.35e3·23-s + 690·25-s + 1.45e3·27-s + 2.11e4·29-s − 3.31e3·31-s − 1.62e3·33-s − 2.10e3·37-s + 2.76e4·39-s − 2.53e3·41-s − 1.15e4·43-s − 486·45-s + 1.56e4·47-s − 3.46e4·51-s − 1.65e4·53-s − 540·55-s − 4.04e4·57-s − 1.31e4·59-s − 5.79e3·61-s + 9.21e3·65-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.107·5-s + 1/3·9-s + 0.224·11-s − 2.52·13-s + 0.123·15-s + 1.61·17-s + 1.42·19-s − 2.50·23-s + 0.220·25-s + 0.384·27-s + 4.66·29-s − 0.618·31-s − 0.258·33-s − 0.252·37-s + 2.91·39-s − 0.235·41-s − 0.951·43-s − 0.0357·45-s + 1.03·47-s − 1.86·51-s − 0.807·53-s − 0.0240·55-s − 1.64·57-s − 0.491·59-s − 0.199·61-s + 0.270·65-s + ⋯

Functional equation

Λ(s)=((283478)s/2ΓC(s)4L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((283478)s/2ΓC(s+5/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 2834782^{8} \cdot 3^{4} \cdot 7^{8}
Sign: 11
Analytic conductor: 7.90954×1077.90954\times 10^{7}
Root analytic conductor: 9.711119.71111
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 283478, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{8} \cdot 3^{4} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) \approx 0.59794988030.5979498803
L(12)L(\frac12) \approx 0.59794988030.5979498803
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
3C2C_2 (1+p2T+p4T2)2 ( 1 + p^{2} T + p^{4} T^{2} )^{2}
7 1 1
good5D4×C2D_4\times C_2 1+6T654T26672pT3376081p2T46672p6T5654p10T6+6p15T7+p20T8 1 + 6 T - 654 T^{2} - 6672 p T^{3} - 376081 p^{2} T^{4} - 6672 p^{6} T^{5} - 654 p^{10} T^{6} + 6 p^{15} T^{7} + p^{20} T^{8}
11D4×C2D_4\times C_2 190T43146T2+24377040T324615785185T4+24377040p5T543146p10T690p15T7+p20T8 1 - 90 T - 43146 T^{2} + 24377040 T^{3} - 24615785185 T^{4} + 24377040 p^{5} T^{5} - 43146 p^{10} T^{6} - 90 p^{15} T^{7} + p^{20} T^{8}
13D4D_{4} (1+768T+689558T2+768p5T3+p10T4)2 ( 1 + 768 T + 689558 T^{2} + 768 p^{5} T^{3} + p^{10} T^{4} )^{2}
17D4×C2D_4\times C_2 11926T52038T21775386800T3+6866068206815T41775386800p5T552038p10T61926p15T7+p20T8 1 - 1926 T - 52038 T^{2} - 1775386800 T^{3} + 6866068206815 T^{4} - 1775386800 p^{5} T^{5} - 52038 p^{10} T^{6} - 1926 p^{15} T^{7} + p^{20} T^{8}
19D4×C2D_4\times C_2 12248T+2045674T2+4370939264T39597001149797T4+4370939264p5T5+2045674p10T62248p15T7+p20T8 1 - 2248 T + 2045674 T^{2} + 4370939264 T^{3} - 9597001149797 T^{4} + 4370939264 p^{5} T^{5} + 2045674 p^{10} T^{6} - 2248 p^{15} T^{7} + p^{20} T^{8}
23D4×C2D_4\times C_2 1+6354T+17412870T2+64097627040T3+225899059481879T4+64097627040p5T5+17412870p10T6+6354p15T7+p20T8 1 + 6354 T + 17412870 T^{2} + 64097627040 T^{3} + 225899059481879 T^{4} + 64097627040 p^{5} T^{5} + 17412870 p^{10} T^{6} + 6354 p^{15} T^{7} + p^{20} T^{8}
29D4D_{4} (110572T+60053694T210572p5T3+p10T4)2 ( 1 - 10572 T + 60053694 T^{2} - 10572 p^{5} T^{3} + p^{10} T^{4} )^{2}
31D4×C2D_4\times C_2 1+3312T20161598T286533816320T3164535742920381T486533816320p5T520161598p10T6+3312p15T7+p20T8 1 + 3312 T - 20161598 T^{2} - 86533816320 T^{3} - 164535742920381 T^{4} - 86533816320 p^{5} T^{5} - 20161598 p^{10} T^{6} + 3312 p^{15} T^{7} + p^{20} T^{8}
37D4×C2D_4\times C_2 1+2104T10065302T2261307954784T34905535826642837T4261307954784p5T510065302p10T6+2104p15T7+p20T8 1 + 2104 T - 10065302 T^{2} - 261307954784 T^{3} - 4905535826642837 T^{4} - 261307954784 p^{5} T^{5} - 10065302 p^{10} T^{6} + 2104 p^{15} T^{7} + p^{20} T^{8}
41D4D_{4} (1+1266T+226048450T2+1266p5T3+p10T4)2 ( 1 + 1266 T + 226048450 T^{2} + 1266 p^{5} T^{3} + p^{10} T^{4} )^{2}
43D4D_{4} (1+5768T18440058T2+5768p5T3+p10T4)2 ( 1 + 5768 T - 18440058 T^{2} + 5768 p^{5} T^{3} + p^{10} T^{4} )^{2}
47D4×C2D_4\times C_2 115612T129736270T2+1330442150400T3+30982015974170739T4+1330442150400p5T5129736270p10T615612p15T7+p20T8 1 - 15612 T - 129736270 T^{2} + 1330442150400 T^{3} + 30982015974170739 T^{4} + 1330442150400 p^{5} T^{5} - 129736270 p^{10} T^{6} - 15612 p^{15} T^{7} + p^{20} T^{8}
53D4×C2D_4\times C_2 1+16512T326970214T23909622657536T3+70632917811042123T43909622657536p5T5326970214p10T6+16512p15T7+p20T8 1 + 16512 T - 326970214 T^{2} - 3909622657536 T^{3} + 70632917811042123 T^{4} - 3909622657536 p^{5} T^{5} - 326970214 p^{10} T^{6} + 16512 p^{15} T^{7} + p^{20} T^{8}
59D4×C2D_4\times C_2 1+13140T369558p2T2+384245136000T3+1494391243648374203T4+384245136000p5T5369558p12T6+13140p15T7+p20T8 1 + 13140 T - 369558 p^{2} T^{2} + 384245136000 T^{3} + 1494391243648374203 T^{4} + 384245136000 p^{5} T^{5} - 369558 p^{12} T^{6} + 13140 p^{15} T^{7} + p^{20} T^{8}
61D4×C2D_4\times C_2 1+5796T1275860366T22200965041520T3+972953775501959307T42200965041520p5T51275860366p10T6+5796p15T7+p20T8 1 + 5796 T - 1275860366 T^{2} - 2200965041520 T^{3} + 972953775501959307 T^{4} - 2200965041520 p^{5} T^{5} - 1275860366 p^{10} T^{6} + 5796 p^{15} T^{7} + p^{20} T^{8}
67D4×C2D_4\times C_2 156116T+850173514T2+22525987751552T3789760835217853877T4+22525987751552p5T5+850173514p10T656116p15T7+p20T8 1 - 56116 T + 850173514 T^{2} + 22525987751552 T^{3} - 789760835217853877 T^{4} + 22525987751552 p^{5} T^{5} + 850173514 p^{10} T^{6} - 56116 p^{15} T^{7} + p^{20} T^{8}
71D4D_{4} (111022T822078402T211022p5T3+p10T4)2 ( 1 - 11022 T - 822078402 T^{2} - 11022 p^{5} T^{3} + p^{10} T^{4} )^{2}
73D4×C2D_4\times C_2 1+85384T+1490284450T2+141225120630880T3+14230455000486300979T4+141225120630880p5T5+1490284450p10T6+85384p15T7+p20T8 1 + 85384 T + 1490284450 T^{2} + 141225120630880 T^{3} + 14230455000486300979 T^{4} + 141225120630880 p^{5} T^{5} + 1490284450 p^{10} T^{6} + 85384 p^{15} T^{7} + p^{20} T^{8}
79D4×C2D_4\times C_2 119620T5831522702T21223391444480T3+27991691518342215603T41223391444480p5T55831522702p10T619620p15T7+p20T8 1 - 19620 T - 5831522702 T^{2} - 1223391444480 T^{3} + 27991691518342215603 T^{4} - 1223391444480 p^{5} T^{5} - 5831522702 p^{10} T^{6} - 19620 p^{15} T^{7} + p^{20} T^{8}
83D4D_{4} (144424T+7801188630T244424p5T3+p10T4)2 ( 1 - 44424 T + 7801188630 T^{2} - 44424 p^{5} T^{3} + p^{10} T^{4} )^{2}
89D4×C2D_4\times C_2 1211218T+23178254114T22168505612203616T3+ 1 - 211218 T + 23178254114 T^{2} - 2168505612203616 T^{3} + 17 ⁣ ⁣2317\!\cdots\!23T42168505612203616p5T5+23178254114p10T6211218p15T7+p20T8 T^{4} - 2168505612203616 p^{5} T^{5} + 23178254114 p^{10} T^{6} - 211218 p^{15} T^{7} + p^{20} T^{8}
97D4D_{4} (1+44864T+3170652414T2+44864p5T3+p10T4)2 ( 1 + 44864 T + 3170652414 T^{2} + 44864 p^{5} T^{3} + p^{10} T^{4} )^{2}
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   L(s)=p j=18(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−6.88635258917440491606022157397, −6.56699996784460935574517593423, −6.39416791247236457151031355143, −6.21697453981815996563972831298, −5.89751611567900320596305456292, −5.78060334620223628473196694089, −5.21062515158322478743735948884, −5.19916182954355879991555913375, −5.06077940167807153813253860129, −4.71910719493133869426373524742, −4.64378193596545868062640878542, −4.29943828324176421824393290194, −3.82673394934010298392069978200, −3.64750509943567458252656660484, −3.33629306565436695452286483610, −3.02498987225598824498461885088, −2.62448035244952068694321690546, −2.49632413113200367686310444966, −2.33762802900758455706464986772, −1.59804112156089165665430964074, −1.55446426082211897692312589327, −1.02062251940713316132239054433, −0.799937453075725890137391300786, −0.52645699331340661634076803105, −0.11862014060307787771893686478, 0.11862014060307787771893686478, 0.52645699331340661634076803105, 0.799937453075725890137391300786, 1.02062251940713316132239054433, 1.55446426082211897692312589327, 1.59804112156089165665430964074, 2.33762802900758455706464986772, 2.49632413113200367686310444966, 2.62448035244952068694321690546, 3.02498987225598824498461885088, 3.33629306565436695452286483610, 3.64750509943567458252656660484, 3.82673394934010298392069978200, 4.29943828324176421824393290194, 4.64378193596545868062640878542, 4.71910719493133869426373524742, 5.06077940167807153813253860129, 5.19916182954355879991555913375, 5.21062515158322478743735948884, 5.78060334620223628473196694089, 5.89751611567900320596305456292, 6.21697453981815996563972831298, 6.39416791247236457151031355143, 6.56699996784460935574517593423, 6.88635258917440491606022157397

Graph of the ZZ-function along the critical line