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Properties

Label 8-6480e4-1.1-c1e4-0-3
Degree 88
Conductor 1.763×10151.763\times 10^{15}
Sign 11
Analytic cond. 7.16817×1067.16817\times 10^{6}
Root an. cond. 7.193267.19326
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 44

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 2·7-s − 4·11-s − 2·17-s − 10·23-s + 10·25-s − 4·29-s − 14·31-s − 8·35-s + 14·37-s + 4·41-s − 22·43-s − 7·49-s − 8·53-s − 16·55-s + 4·61-s − 24·67-s − 28·71-s + 2·73-s + 8·77-s − 16·79-s − 6·83-s − 8·85-s − 8·89-s − 10·97-s − 8·101-s + 4·103-s + ⋯
L(s)  = 1  + 1.78·5-s − 0.755·7-s − 1.20·11-s − 0.485·17-s − 2.08·23-s + 2·25-s − 0.742·29-s − 2.51·31-s − 1.35·35-s + 2.30·37-s + 0.624·41-s − 3.35·43-s − 49-s − 1.09·53-s − 2.15·55-s + 0.512·61-s − 2.93·67-s − 3.32·71-s + 0.234·73-s + 0.911·77-s − 1.80·79-s − 0.658·83-s − 0.867·85-s − 0.847·89-s − 1.01·97-s − 0.796·101-s + 0.394·103-s + ⋯

Functional equation

Λ(s)=((216⋅316⋅54)s/2 ΓC(s)4 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((216⋅316⋅54)s/2 ΓC(s+1/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 216⋅316⋅542^{16} \cdot 3^{16} \cdot 5^{4}
Sign: 11
Analytic conductor: 7.16817×1067.16817\times 10^{6}
Root analytic conductor: 7.193267.19326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 216⋅316⋅54, ( :1/2,1/2,1/2,1/2), 1)(8,\ 2^{16} \cdot 3^{16} \cdot 5^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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)
bad2 1 1
3 1 1
5C1C_1 (1−T)4 ( 1 - T )^{4}
good7C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2T+11T2+38T4+11p2T6+2p3T7+p4T8 1 + 2 T + 11 T^{2} + 38 T^{4} + 11 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
11C2≀C2≀C2C_2 \wr C_2\wr C_2 1+4T+9T2−12T3−152T4−12pT5+9p2T6+4p3T7+p4T8 1 + 4 T + 9 T^{2} - 12 T^{3} - 152 T^{4} - 12 p T^{5} + 9 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
13C2≀C2≀C2C_2 \wr C_2\wr C_2 1+pT2+36T3+216T4+36pT5+p3T6+p4T8 1 + p T^{2} + 36 T^{3} + 216 T^{4} + 36 p T^{5} + p^{3} T^{6} + p^{4} T^{8}
17C2≀C2≀C2C_2 \wr C_2\wr C_2 1+2T+30T2−18T3+370T4−18pT5+30p2T6+2p3T7+p4T8 1 + 2 T + 30 T^{2} - 18 T^{3} + 370 T^{4} - 18 p T^{5} + 30 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}
19C22≀C2C_2^2 \wr C_2 1+26T2+699T4+26p2T6+p4T8 1 + 26 T^{2} + 699 T^{4} + 26 p^{2} T^{6} + p^{4} T^{8}
23C2≀C2≀C2C_2 \wr C_2\wr C_2 1+10T+117T2+30pT3+4312T4+30p2T5+117p2T6+10p3T7+p4T8 1 + 10 T + 117 T^{2} + 30 p T^{3} + 4312 T^{4} + 30 p^{2} T^{5} + 117 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8}
29C2≀C2≀C2C_2 \wr C_2\wr C_2 1+4T+69T2+444T3+2272T4+444pT5+69p2T6+4p3T7+p4T8 1 + 4 T + 69 T^{2} + 444 T^{3} + 2272 T^{4} + 444 p T^{5} + 69 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}
31C2≀C2≀C2C_2 \wr C_2\wr C_2 1+14T+185T2+1386T3+9572T4+1386pT5+185p2T6+14p3T7+p4T8 1 + 14 T + 185 T^{2} + 1386 T^{3} + 9572 T^{4} + 1386 p T^{5} + 185 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8}
37C2≀C2≀C2C_2 \wr C_2\wr C_2 1−14T+146T2−1074T3+6914T4−1074pT5+146p2T6−14p3T7+p4T8 1 - 14 T + 146 T^{2} - 1074 T^{3} + 6914 T^{4} - 1074 p T^{5} + 146 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8}
41C2≀C2≀C2C_2 \wr C_2\wr C_2 1−4T+42T2+192T3+151T4+192pT5+42p2T6−4p3T7+p4T8 1 - 4 T + 42 T^{2} + 192 T^{3} + 151 T^{4} + 192 p T^{5} + 42 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
43C2≀C2≀C2C_2 \wr C_2\wr C_2 1+22T+242T2+1662T3+10346T4+1662pT5+242p2T6+22p3T7+p4T8 1 + 22 T + 242 T^{2} + 1662 T^{3} + 10346 T^{4} + 1662 p T^{5} + 242 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8}
47C2≀C2≀C2C_2 \wr C_2\wr C_2 1+75T2−462T3+2558T4−462pT5+75p2T6+p4T8 1 + 75 T^{2} - 462 T^{3} + 2558 T^{4} - 462 p T^{5} + 75 p^{2} T^{6} + p^{4} T^{8}
53C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+75T2+66T3+1834T4+66pT5+75p2T6+8p3T7+p4T8 1 + 8 T + 75 T^{2} + 66 T^{3} + 1834 T^{4} + 66 p T^{5} + 75 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
59C22C_2^2 (1+115T2+p2T4)2 ( 1 + 115 T^{2} + p^{2} T^{4} )^{2}
61C2≀C2≀C2C_2 \wr C_2\wr C_2 1−4T+92T2+228T3+2630T4+228pT5+92p2T6−4p3T7+p4T8 1 - 4 T + 92 T^{2} + 228 T^{3} + 2630 T^{4} + 228 p T^{5} + 92 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}
67C2≀C2≀C2C_2 \wr C_2\wr C_2 1+24T+272T2+1560T3+8526T4+1560pT5+272p2T6+24p3T7+p4T8 1 + 24 T + 272 T^{2} + 1560 T^{3} + 8526 T^{4} + 1560 p T^{5} + 272 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8}
71C2≀C2≀C2C_2 \wr C_2\wr C_2 1+28T+405T2+3744T3+31000T4+3744pT5+405p2T6+28p3T7+p4T8 1 + 28 T + 405 T^{2} + 3744 T^{3} + 31000 T^{4} + 3744 p T^{5} + 405 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8}
73C2≀C2≀C2C_2 \wr C_2\wr C_2 1−2T+206T2+18T3+18866T4+18pT5+206p2T6−2p3T7+p4T8 1 - 2 T + 206 T^{2} + 18 T^{3} + 18866 T^{4} + 18 p T^{5} + 206 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8}
79D4D_{4} (1+8T+162T2+8pT3+p2T4)2 ( 1 + 8 T + 162 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2}
83C2≀C2≀C2C_2 \wr C_2\wr C_2 1+6T+254T2+990T3+28314T4+990pT5+254p2T6+6p3T7+p4T8 1 + 6 T + 254 T^{2} + 990 T^{3} + 28314 T^{4} + 990 p T^{5} + 254 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}
89C2≀C2≀C2C_2 \wr C_2\wr C_2 1+8T+pT2+1472T3+11344T4+1472pT5+p3T6+8p3T7+p4T8 1 + 8 T + p T^{2} + 1472 T^{3} + 11344 T^{4} + 1472 p T^{5} + p^{3} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8}
97C2≀C2≀C2C_2 \wr C_2\wr C_2 1+10T+274T2+2782T3+34306T4+2782pT5+274p2T6+10p3T7+p4T8 1 + 10 T + 274 T^{2} + 2782 T^{3} + 34306 T^{4} + 2782 p T^{5} + 274 p^{2} T^{6} + 10 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

−6.04375876223917170312876983222, −5.85242828563078944010608264534, −5.67021972017690320419486452180, −5.44978277006977802115247554048, −5.40295917157974544089985171397, −5.16637573268202200775598186736, −4.78682245891604885499481735250, −4.75604007723863867480755006877, −4.60417555476207775482409689858, −4.38530956867976820543473215436, −4.02514543726491529826392282211, −3.94576574135476064578232111294, −3.75655709666117738456816164020, −3.28179441814298675613681438978, −3.20121127914915394671140853899, −3.18717706319921795122653278243, −2.93245470216733428765192393534, −2.36386119914992939268171827656, −2.31621785707377617863700511445, −2.29730759574997686860877237100, −2.23342155993214923330526509632, −1.52369230532649357921893637208, −1.36729317802085522659371928118, −1.35219283967426282493437666435, −1.30724537858225723440153223838, 0, 0, 0, 0, 1.30724537858225723440153223838, 1.35219283967426282493437666435, 1.36729317802085522659371928118, 1.52369230532649357921893637208, 2.23342155993214923330526509632, 2.29730759574997686860877237100, 2.31621785707377617863700511445, 2.36386119914992939268171827656, 2.93245470216733428765192393534, 3.18717706319921795122653278243, 3.20121127914915394671140853899, 3.28179441814298675613681438978, 3.75655709666117738456816164020, 3.94576574135476064578232111294, 4.02514543726491529826392282211, 4.38530956867976820543473215436, 4.60417555476207775482409689858, 4.75604007723863867480755006877, 4.78682245891604885499481735250, 5.16637573268202200775598186736, 5.40295917157974544089985171397, 5.44978277006977802115247554048, 5.67021972017690320419486452180, 5.85242828563078944010608264534, 6.04375876223917170312876983222

Graph of the ZZ-function along the critical line