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Properties

Label 8-5e16-1.1-c1e4-0-9
Degree 88
Conductor 152587890625152587890625
Sign 11
Analytic cond. 620.338620.338
Root an. cond. 2.233972.23397
Motivic weight 11
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  + 3·2-s + 4·3-s + 7·4-s + 12·6-s − 12·7-s + 15·8-s + 13·9-s + 3·11-s + 28·12-s + 9·13-s − 36·14-s + 30·16-s + 3·17-s + 39·18-s − 48·21-s + 9·22-s + 4·23-s + 60·24-s + 27·26-s + 30·27-s − 84·28-s + 15·29-s + 13·31-s + 57·32-s + 12·33-s + 9·34-s + 91·36-s + ⋯
L(s)  = 1  + 2.12·2-s + 2.30·3-s + 7/2·4-s + 4.89·6-s − 4.53·7-s + 5.30·8-s + 13/3·9-s + 0.904·11-s + 8.08·12-s + 2.49·13-s − 9.62·14-s + 15/2·16-s + 0.727·17-s + 9.19·18-s − 10.4·21-s + 1.91·22-s + 0.834·23-s + 12.2·24-s + 5.29·26-s + 5.77·27-s − 15.8·28-s + 2.78·29-s + 2.33·31-s + 10.0·32-s + 2.08·33-s + 1.54·34-s + 91/6·36-s + ⋯

Functional equation

Λ(s)=((516)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((516)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 5165^{16}
Sign: 11
Analytic conductor: 620.338620.338
Root analytic conductor: 2.233972.23397
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 516, ( :1/2,1/2,1/2,1/2), 1)(8,\ 5^{16} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 33.9364376533.93643765
L(12)L(\frac12) ≈\approx 33.9364376533.93643765
L(32)L(\frac{3}{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)
bad5 1 1
good2C22:C4C_2^2:C_4 1−3T+pT2+T4+p3T6−3p3T7+p4T8 1 - 3 T + p T^{2} + T^{4} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
3C22:C4C_2^2:C_4 1−4T+pT2+10T3−29T4+10pT5+p3T6−4p3T7+p4T8 1 - 4 T + p T^{2} + 10 T^{3} - 29 T^{4} + 10 p T^{5} + p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
7C2C_2 (1+3T+pT2)4 ( 1 + 3 T + p T^{2} )^{4}
11C4×C2C_4\times C_2 1−3T−2T2+39T3−95T4+39pT5−2p2T6−3p3T7+p4T8 1 - 3 T - 2 T^{2} + 39 T^{3} - 95 T^{4} + 39 p T^{5} - 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
13C22:C4C_2^2:C_4 1−9T+23T2−15T3+16T4−15pT5+23p2T6−9p3T7+p4T8 1 - 9 T + 23 T^{2} - 15 T^{3} + 16 T^{4} - 15 p T^{5} + 23 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}
17C22:C4C_2^2:C_4 1−3T+2T2−75T3+511T4−75pT5+2p2T6−3p3T7+p4T8 1 - 3 T + 2 T^{2} - 75 T^{3} + 511 T^{4} - 75 p T^{5} + 2 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
19C22:C4C_2^2:C_4 1−9T2+70T3+291T4+70pT5−9p2T6+p4T8 1 - 9 T^{2} + 70 T^{3} + 291 T^{4} + 70 p T^{5} - 9 p^{2} T^{6} + p^{4} T^{8}
23C22:C4C_2^2:C_4 1−4T−7T2−70T3+821T4−70pT5−7p2T6−4p3T7+p4T8 1 - 4 T - 7 T^{2} - 70 T^{3} + 821 T^{4} - 70 p T^{5} - 7 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
29C4×C2C_4\times C_2 1−15T+106T2−675T3+4171T4−675pT5+106p2T6−15p3T7+p4T8 1 - 15 T + 106 T^{2} - 675 T^{3} + 4171 T^{4} - 675 p T^{5} + 106 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8}
31C22:C4C_2^2:C_4 1−13T+63T2−11pT3+80pT4−11p2T5+63p2T6−13p3T7+p4T8 1 - 13 T + 63 T^{2} - 11 p T^{3} + 80 p T^{4} - 11 p^{2} T^{5} + 63 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}
37C22:C4C_2^2:C_4 1+12T+107T2+870T3+6661T4+870pT5+107p2T6+12p3T7+p4T8 1 + 12 T + 107 T^{2} + 870 T^{3} + 6661 T^{4} + 870 p T^{5} + 107 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}
41C4×C2C_4\times C_2 1−3T−32T2+219T3+655T4+219pT5−32p2T6−3p3T7+p4T8 1 - 3 T - 32 T^{2} + 219 T^{3} + 655 T^{4} + 219 p T^{5} - 32 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
43C2C_2 (1+9T+pT2)4 ( 1 + 9 T + p T^{2} )^{4}
47C22:C4C_2^2:C_4 1+17T+137T2+1145T3+9596T4+1145pT5+137p2T6+17p3T7+p4T8 1 + 17 T + 137 T^{2} + 1145 T^{3} + 9596 T^{4} + 1145 p T^{5} + 137 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8}
53C22:C4C_2^2:C_4 1+T−37T2−305T3+1976T4−305pT5−37p2T6+p3T7+p4T8 1 + T - 37 T^{2} - 305 T^{3} + 1976 T^{4} - 305 p T^{5} - 37 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}
59C22:C4C_2^2:C_4 1+31T2−210T3+2851T4−210pT5+31p2T6+p4T8 1 + 31 T^{2} - 210 T^{3} + 2851 T^{4} - 210 p T^{5} + 31 p^{2} T^{6} + p^{4} T^{8}
61C4C_4×\timesC4C_4 (1−29T+331T2−29pT3+p2T4)(1+16T+166T2+16pT3+p2T4) ( 1 - 29 T + 331 T^{2} - 29 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 166 T^{2} + 16 p T^{3} + p^{2} T^{4} )
67C22:C4C_2^2:C_4 1−3T+77T2−375T3+3436T4−375pT5+77p2T6−3p3T7+p4T8 1 - 3 T + 77 T^{2} - 375 T^{3} + 3436 T^{4} - 375 p T^{5} + 77 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
71C4×C2C_4\times C_2 1−3T−62T2+399T3+3205T4+399pT5−62p2T6−3p3T7+p4T8 1 - 3 T - 62 T^{2} + 399 T^{3} + 3205 T^{4} + 399 p T^{5} - 62 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
73C22:C4C_2^2:C_4 1+6T−37T2+300T3+7381T4+300pT5−37p2T6+6p3T7+p4T8 1 + 6 T - 37 T^{2} + 300 T^{3} + 7381 T^{4} + 300 p T^{5} - 37 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
79C22:C4C_2^2:C_4 1+15T+6T2−815T3−5979T4−815pT5+6p2T6+15p3T7+p4T8 1 + 15 T + 6 T^{2} - 815 T^{3} - 5979 T^{4} - 815 p T^{5} + 6 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8}
83C22:C4C_2^2:C_4 1−14T−7T2+640T3−2059T4+640pT5−7p2T6−14p3T7+p4T8 1 - 14 T - 7 T^{2} + 640 T^{3} - 2059 T^{4} + 640 p T^{5} - 7 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
89C4×C2C_4\times C_2 1−30T+451T2−5400T3+56341T4−5400pT5+451p2T6−30p3T7+p4T8 1 - 30 T + 451 T^{2} - 5400 T^{3} + 56341 T^{4} - 5400 p T^{5} + 451 p^{2} T^{6} - 30 p^{3} T^{7} + p^{4} T^{8}
97C22:C4C_2^2:C_4 1−3T−43T2−765T3+11236T4−765pT5−43p2T6−3p3T7+p4T8 1 - 3 T - 43 T^{2} - 765 T^{3} + 11236 T^{4} - 765 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}
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   L(s)=∏p ∏j=18(1−αj,p p−s)−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

−7.50684870469363063174175620980, −7.22381330968840907906094937720, −6.74456190970950968403419079276, −6.71130919378217510094103776679, −6.70686955884954272532537570585, −6.48111086092776349042563356192, −6.41145826605197682644871307941, −6.21748982557253763632163374867, −6.06711439472992073531994287364, −5.20585015792649268876063448044, −5.20137392361901889397841789226, −4.76201622512382784594372815926, −4.73386085956054442173962955082, −3.97619891696898043366659814084, −3.91238675239480184149859892589, −3.79123734299926871180519417573, −3.32049576332143229011766960735, −3.30873241719855616087609529590, −3.17431869568649634579266873876, −3.04827072211827463998116430552, −2.76177653555214641126305964775, −2.17891869295448590502724055618, −1.69067021465241943755698994487, −1.28017460048614906423799288955, −1.12748878967438659210837673130, 1.12748878967438659210837673130, 1.28017460048614906423799288955, 1.69067021465241943755698994487, 2.17891869295448590502724055618, 2.76177653555214641126305964775, 3.04827072211827463998116430552, 3.17431869568649634579266873876, 3.30873241719855616087609529590, 3.32049576332143229011766960735, 3.79123734299926871180519417573, 3.91238675239480184149859892589, 3.97619891696898043366659814084, 4.73386085956054442173962955082, 4.76201622512382784594372815926, 5.20137392361901889397841789226, 5.20585015792649268876063448044, 6.06711439472992073531994287364, 6.21748982557253763632163374867, 6.41145826605197682644871307941, 6.48111086092776349042563356192, 6.70686955884954272532537570585, 6.71130919378217510094103776679, 6.74456190970950968403419079276, 7.22381330968840907906094937720, 7.50684870469363063174175620980

Graph of the ZZ-function along the critical line