Properties

Label 8-832e4-1.1-c5e4-0-0
Degree 88
Conductor 479174066176479174066176
Sign 11
Analytic cond. 3.17055×1083.17055\times 10^{8}
Root an. cond. 11.551511.5515
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  + 11·3-s − 31·5-s + 39·7-s − 418·9-s − 340·11-s + 676·13-s − 341·15-s + 1.75e3·17-s − 2.00e3·19-s + 429·21-s + 6.79e3·23-s − 4.19e3·25-s − 9.71e3·27-s − 6.76e3·29-s + 1.86e4·31-s − 3.74e3·33-s − 1.20e3·35-s + 573·37-s + 7.43e3·39-s + 1.89e3·41-s − 2.83e4·43-s + 1.29e4·45-s + 2.16e4·47-s − 3.55e4·49-s + 1.93e4·51-s + 4.59e4·53-s + 1.05e4·55-s + ⋯
L(s)  = 1  + 0.705·3-s − 0.554·5-s + 0.300·7-s − 1.72·9-s − 0.847·11-s + 1.10·13-s − 0.391·15-s + 1.47·17-s − 1.27·19-s + 0.212·21-s + 2.67·23-s − 1.34·25-s − 2.56·27-s − 1.49·29-s + 3.48·31-s − 0.597·33-s − 0.166·35-s + 0.0688·37-s + 0.782·39-s + 0.175·41-s − 2.33·43-s + 0.953·45-s + 1.42·47-s − 2.11·49-s + 1.04·51-s + 2.24·53-s + 0.469·55-s + ⋯

Functional equation

Λ(s)=((224134)s/2ΓC(s)4L(s)=(Λ(6s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}
Λ(s)=((224134)s/2ΓC(s+5/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 2241342^{24} \cdot 13^{4}
Sign: 11
Analytic conductor: 3.17055×1083.17055\times 10^{8}
Root analytic conductor: 11.551511.5515
Motivic weight: 55
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (8, 224134, ( :5/2,5/2,5/2,5/2), 1)(8,\ 2^{24} \cdot 13^{4} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )

Particular Values

L(3)L(3) \approx 6.7117080246.711708024
L(12)L(\frac12) \approx 6.7117080246.711708024
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
13C1C_1 (1p2T)4 ( 1 - p^{2} T )^{4}
good3C2S4C_2 \wr S_4 111T+539T2808T3+39808pT4808p5T5+539p10T611p15T7+p20T8 1 - 11 T + 539 T^{2} - 808 T^{3} + 39808 p T^{4} - 808 p^{5} T^{5} + 539 p^{10} T^{6} - 11 p^{15} T^{7} + p^{20} T^{8}
5C2S4C_2 \wr S_4 1+31T+5153T2+252422T3+22858118T4+252422p5T5+5153p10T6+31p15T7+p20T8 1 + 31 T + 5153 T^{2} + 252422 T^{3} + 22858118 T^{4} + 252422 p^{5} T^{5} + 5153 p^{10} T^{6} + 31 p^{15} T^{7} + p^{20} T^{8}
7C2S4C_2 \wr S_4 139T+37057T2+1509244T3+574546398T4+1509244p5T5+37057p10T639p15T7+p20T8 1 - 39 T + 37057 T^{2} + 1509244 T^{3} + 574546398 T^{4} + 1509244 p^{5} T^{5} + 37057 p^{10} T^{6} - 39 p^{15} T^{7} + p^{20} T^{8}
11C2S4C_2 \wr S_4 1+340T+127772T23699508pT32539680526pT43699508p6T5+127772p10T6+340p15T7+p20T8 1 + 340 T + 127772 T^{2} - 3699508 p T^{3} - 2539680526 p T^{4} - 3699508 p^{6} T^{5} + 127772 p^{10} T^{6} + 340 p^{15} T^{7} + p^{20} T^{8}
17C2S4C_2 \wr S_4 11757T+3521321T24384174254T3+7201578383694T44384174254p5T5+3521321p10T61757p15T7+p20T8 1 - 1757 T + 3521321 T^{2} - 4384174254 T^{3} + 7201578383694 T^{4} - 4384174254 p^{5} T^{5} + 3521321 p^{10} T^{6} - 1757 p^{15} T^{7} + p^{20} T^{8}
19C2S4C_2 \wr S_4 1+2000T+1315204T2+5123411728T3+16378456307222T4+5123411728p5T5+1315204p10T6+2000p15T7+p20T8 1 + 2000 T + 1315204 T^{2} + 5123411728 T^{3} + 16378456307222 T^{4} + 5123411728 p^{5} T^{5} + 1315204 p^{10} T^{6} + 2000 p^{15} T^{7} + p^{20} T^{8}
23C2S4C_2 \wr S_4 16796T+37313132T2124874503804T3+376774335025606T4124874503804p5T5+37313132p10T66796p15T7+p20T8 1 - 6796 T + 37313132 T^{2} - 124874503804 T^{3} + 376774335025606 T^{4} - 124874503804 p^{5} T^{5} + 37313132 p^{10} T^{6} - 6796 p^{15} T^{7} + p^{20} T^{8}
29C2S4C_2 \wr S_4 1+6768T+44087996T2+145395725328T3+820460284261782T4+145395725328p5T5+44087996p10T6+6768p15T7+p20T8 1 + 6768 T + 44087996 T^{2} + 145395725328 T^{3} + 820460284261782 T^{4} + 145395725328 p^{5} T^{5} + 44087996 p^{10} T^{6} + 6768 p^{15} T^{7} + p^{20} T^{8}
31C2S4C_2 \wr S_4 118638T+155809588T2751376049398T3+3287012745889942T4751376049398p5T5+155809588p10T618638p15T7+p20T8 1 - 18638 T + 155809588 T^{2} - 751376049398 T^{3} + 3287012745889942 T^{4} - 751376049398 p^{5} T^{5} + 155809588 p^{10} T^{6} - 18638 p^{15} T^{7} + p^{20} T^{8}
37C2S4C_2 \wr S_4 1573T+239640937T2202243860778T3+23587529015810070T4202243860778p5T5+239640937p10T6573p15T7+p20T8 1 - 573 T + 239640937 T^{2} - 202243860778 T^{3} + 23587529015810070 T^{4} - 202243860778 p^{5} T^{5} + 239640937 p^{10} T^{6} - 573 p^{15} T^{7} + p^{20} T^{8}
41C2S4C_2 \wr S_4 11890T+311989748T2958041784790T3+46445767927334790T4958041784790p5T5+311989748p10T61890p15T7+p20T8 1 - 1890 T + 311989748 T^{2} - 958041784790 T^{3} + 46445767927334790 T^{4} - 958041784790 p^{5} T^{5} + 311989748 p^{10} T^{6} - 1890 p^{15} T^{7} + p^{20} T^{8}
43C2S4C_2 \wr S_4 1+28327T+616834087T2+7694590240592T3+104664597506136140T4+7694590240592p5T5+616834087p10T6+28327p15T7+p20T8 1 + 28327 T + 616834087 T^{2} + 7694590240592 T^{3} + 104664597506136140 T^{4} + 7694590240592 p^{5} T^{5} + 616834087 p^{10} T^{6} + 28327 p^{15} T^{7} + p^{20} T^{8}
47C2S4C_2 \wr S_4 121603T+666176561T210471292070516T3+201019507301163414T410471292070516p5T5+666176561p10T621603p15T7+p20T8 1 - 21603 T + 666176561 T^{2} - 10471292070516 T^{3} + 201019507301163414 T^{4} - 10471292070516 p^{5} T^{5} + 666176561 p^{10} T^{6} - 21603 p^{15} T^{7} + p^{20} T^{8}
53C2S4C_2 \wr S_4 145950T+1702757516T247289854346690T3+1021571292731542854T447289854346690p5T5+1702757516p10T645950p15T7+p20T8 1 - 45950 T + 1702757516 T^{2} - 47289854346690 T^{3} + 1021571292731542854 T^{4} - 47289854346690 p^{5} T^{5} + 1702757516 p^{10} T^{6} - 45950 p^{15} T^{7} + p^{20} T^{8}
59C2S4C_2 \wr S_4 1+916T+2171389052T28148340134204T3+2041156776912096870T48148340134204p5T5+2171389052p10T6+916p15T7+p20T8 1 + 916 T + 2171389052 T^{2} - 8148340134204 T^{3} + 2041156776912096870 T^{4} - 8148340134204 p^{5} T^{5} + 2171389052 p^{10} T^{6} + 916 p^{15} T^{7} + p^{20} T^{8}
61C2S4C_2 \wr S_4 155730T+67085956pT2139009698415134T3+5445275958006059382T4139009698415134p5T5+67085956p11T655730p15T7+p20T8 1 - 55730 T + 67085956 p T^{2} - 139009698415134 T^{3} + 5445275958006059382 T^{4} - 139009698415134 p^{5} T^{5} + 67085956 p^{11} T^{6} - 55730 p^{15} T^{7} + p^{20} T^{8}
67C2S4C_2 \wr S_4 1+7736T+2054018164T2+34631039929752T3+3527055037627735350T4+34631039929752p5T5+2054018164p10T6+7736p15T7+p20T8 1 + 7736 T + 2054018164 T^{2} + 34631039929752 T^{3} + 3527055037627735350 T^{4} + 34631039929752 p^{5} T^{5} + 2054018164 p^{10} T^{6} + 7736 p^{15} T^{7} + p^{20} T^{8}
71C2S4C_2 \wr S_4 125229T+5569425785T291697921781076T3+13582162611069378350T491697921781076p5T5+5569425785p10T625229p15T7+p20T8 1 - 25229 T + 5569425785 T^{2} - 91697921781076 T^{3} + 13582162611069378350 T^{4} - 91697921781076 p^{5} T^{5} + 5569425785 p^{10} T^{6} - 25229 p^{15} T^{7} + p^{20} T^{8}
73C2S4C_2 \wr S_4 1+82484T+3610412740T227086603654356T34146949427050770410T427086603654356p5T5+3610412740p10T6+82484p15T7+p20T8 1 + 82484 T + 3610412740 T^{2} - 27086603654356 T^{3} - 4146949427050770410 T^{4} - 27086603654356 p^{5} T^{5} + 3610412740 p^{10} T^{6} + 82484 p^{15} T^{7} + p^{20} T^{8}
79C2S4C_2 \wr S_4 14936T+3136547068T2276560357630760T3+2205991352982184134T4276560357630760p5T5+3136547068p10T64936p15T7+p20T8 1 - 4936 T + 3136547068 T^{2} - 276560357630760 T^{3} + 2205991352982184134 T^{4} - 276560357630760 p^{5} T^{5} + 3136547068 p^{10} T^{6} - 4936 p^{15} T^{7} + p^{20} T^{8}
83C2S4C_2 \wr S_4 1119782T+16058583212T21197963265006102T3+91320962773018703350T41197963265006102p5T5+16058583212p10T6119782p15T7+p20T8 1 - 119782 T + 16058583212 T^{2} - 1197963265006102 T^{3} + 91320962773018703350 T^{4} - 1197963265006102 p^{5} T^{5} + 16058583212 p^{10} T^{6} - 119782 p^{15} T^{7} + p^{20} T^{8}
89C2S4C_2 \wr S_4 1+153492T+23233599524T2+2201629256787724T3+ 1 + 153492 T + 23233599524 T^{2} + 2201629256787724 T^{3} + 20 ⁣ ⁣5020\!\cdots\!50T4+2201629256787724p5T5+23233599524p10T6+153492p15T7+p20T8 T^{4} + 2201629256787724 p^{5} T^{5} + 23233599524 p^{10} T^{6} + 153492 p^{15} T^{7} + p^{20} T^{8}
97C2S4C_2 \wr S_4 113364T+15673117972T2+697855089119668T3+ 1 - 13364 T + 15673117972 T^{2} + 697855089119668 T^{3} + 10 ⁣ ⁣5410\!\cdots\!54T4+697855089119668p5T5+15673117972p10T613364p15T7+p20T8 T^{4} + 697855089119668 p^{5} T^{5} + 15673117972 p^{10} T^{6} - 13364 p^{15} T^{7} + p^{20} T^{8}
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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

−6.68452944058160745639535847591, −6.21388352156540350942989652364, −5.96396305695149126701933393067, −5.86004992147183374143238608691, −5.74046864824556768238284674250, −5.31105978721251068369414692196, −5.23585697986123428123098977581, −4.95571247128615953641857023079, −4.77679496844087350546108824602, −4.27494296219800578860774614628, −4.08124675641141145590914122318, −3.90391109438156301003332171714, −3.68053775398656905460239990908, −3.17490520488419122908248086056, −3.08766452296930958295682149124, −2.93715948922570929185706773372, −2.80787044476721731047360195841, −2.33137836983128159142152636132, −1.98851508988812689578286299673, −1.86003840601409194800642001748, −1.42673027344858036112227733189, −1.08720116971035349170994543447, −0.61079487544616993458609765486, −0.54774658151306755564396219249, −0.31418994809951318271021889752, 0.31418994809951318271021889752, 0.54774658151306755564396219249, 0.61079487544616993458609765486, 1.08720116971035349170994543447, 1.42673027344858036112227733189, 1.86003840601409194800642001748, 1.98851508988812689578286299673, 2.33137836983128159142152636132, 2.80787044476721731047360195841, 2.93715948922570929185706773372, 3.08766452296930958295682149124, 3.17490520488419122908248086056, 3.68053775398656905460239990908, 3.90391109438156301003332171714, 4.08124675641141145590914122318, 4.27494296219800578860774614628, 4.77679496844087350546108824602, 4.95571247128615953641857023079, 5.23585697986123428123098977581, 5.31105978721251068369414692196, 5.74046864824556768238284674250, 5.86004992147183374143238608691, 5.96396305695149126701933393067, 6.21388352156540350942989652364, 6.68452944058160745639535847591

Graph of the ZZ-function along the critical line