Properties

Label 8-45e4-1.1-c25e4-0-0
Degree 88
Conductor 41006254100625
Sign 11
Analytic cond. 1.00836×1091.00836\times 10^{9}
Root an. cond. 13.349113.3491
Motivic weight 2525
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  − 7.68e3·2-s − 2.01e7·4-s + 9.76e8·5-s + 2.49e10·7-s + 1.65e11·8-s − 7.50e12·10-s − 9.98e12·11-s − 1.43e14·13-s − 1.91e14·14-s + 1.23e15·16-s − 2.80e15·17-s + 1.50e16·19-s − 1.97e16·20-s + 7.67e16·22-s − 4.87e16·23-s + 5.96e17·25-s + 1.10e18·26-s − 5.02e17·28-s + 3.66e17·29-s + 5.26e18·31-s − 2.14e18·32-s + 2.15e19·34-s + 2.43e19·35-s − 8.25e19·37-s − 1.15e20·38-s + 1.61e20·40-s − 1.16e20·41-s + ⋯
L(s)  = 1  − 1.32·2-s − 0.601·4-s + 1.78·5-s + 0.680·7-s + 0.850·8-s − 2.37·10-s − 0.959·11-s − 1.70·13-s − 0.902·14-s + 1.09·16-s − 1.16·17-s + 1.55·19-s − 1.07·20-s + 1.27·22-s − 0.464·23-s + 2·25-s + 2.26·26-s − 0.409·28-s + 0.192·29-s + 1.20·31-s − 0.328·32-s + 1.54·34-s + 1.21·35-s − 2.06·37-s − 2.06·38-s + 1.52·40-s − 0.804·41-s + ⋯

Functional equation

Λ(s)=(4100625s/2ΓC(s)4L(s)=(Λ(26s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(26-s) \end{aligned}
Λ(s)=(4100625s/2ΓC(s+25/2)4L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+25/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 88
Conductor: 41006254100625    =    38543^{8} \cdot 5^{4}
Sign: 11
Analytic conductor: 1.00836×1091.00836\times 10^{9}
Root analytic conductor: 13.349113.3491
Motivic weight: 2525
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 4100625, ( :25/2,25/2,25/2,25/2), 1)(8,\ 4100625,\ (\ :25/2, 25/2, 25/2, 25/2),\ 1)

Particular Values

L(13)L(13) == 00
L(12)L(\frac12) == 00
L(272)L(\frac{27}{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)
bad3 1 1
5C1C_1 (1p12T)4 ( 1 - p^{12} T )^{4}
good2C2S4C_2 \wr S_4 1+7683T+19803275p2T2+9350685853p6T3+451253267883p13T4+9350685853p31T5+19803275p52T6+7683p75T7+p100T8 1 + 7683 T + 19803275 p^{2} T^{2} + 9350685853 p^{6} T^{3} + 451253267883 p^{13} T^{4} + 9350685853 p^{31} T^{5} + 19803275 p^{52} T^{6} + 7683 p^{75} T^{7} + p^{100} T^{8}
7C2S4C_2 \wr S_4 1508422464p2T+53246328028610032828p2T2 1 - 508422464 p^{2} T + 53246328028610032828 p^{2} T^{2} - 61 ⁣ ⁣8861\!\cdots\!88p5T3+ p^{5} T^{3} + 94 ⁣ ⁣3094\!\cdots\!30p9T4 p^{9} T^{4} - 61 ⁣ ⁣8861\!\cdots\!88p30T5+53246328028610032828p52T6508422464p77T7+p100T8 p^{30} T^{5} + 53246328028610032828 p^{52} T^{6} - 508422464 p^{77} T^{7} + p^{100} T^{8}
11C2S4C_2 \wr S_4 1+9989114586816T+ 1 + 9989114586816 T + 20 ⁣ ⁣1220\!\cdots\!12pT2+ p T^{2} + 67 ⁣ ⁣6067\!\cdots\!60p3T3+ p^{3} T^{3} + 12 ⁣ ⁣8612\!\cdots\!86p5T4+ p^{5} T^{4} + 67 ⁣ ⁣6067\!\cdots\!60p28T5+ p^{28} T^{5} + 20 ⁣ ⁣1220\!\cdots\!12p51T6+9989114586816p75T7+p100T8 p^{51} T^{6} + 9989114586816 p^{75} T^{7} + p^{100} T^{8}
13C2S4C_2 \wr S_4 1+143366199676168T+ 1 + 143366199676168 T + 19 ⁣ ⁣9619\!\cdots\!96p2T2+ p^{2} T^{2} + 17 ⁣ ⁣4417\!\cdots\!44p2T3+ p^{2} T^{3} + 16 ⁣ ⁣1016\!\cdots\!10p3T4+ p^{3} T^{4} + 17 ⁣ ⁣4417\!\cdots\!44p27T5+ p^{27} T^{5} + 19 ⁣ ⁣9619\!\cdots\!96p52T6+143366199676168p75T7+p100T8 p^{52} T^{6} + 143366199676168 p^{75} T^{7} + p^{100} T^{8}
17C2S4C_2 \wr S_4 1+2801218397375112T+ 1 + 2801218397375112 T + 11 ⁣ ⁣3211\!\cdots\!32T2+ T^{2} + 16 ⁣ ⁣8016\!\cdots\!80pT3+ p T^{3} + 35 ⁣ ⁣5435\!\cdots\!54p2T4+ p^{2} T^{4} + 16 ⁣ ⁣8016\!\cdots\!80p26T5+ p^{26} T^{5} + 11 ⁣ ⁣3211\!\cdots\!32p50T6+2801218397375112p75T7+p100T8 p^{50} T^{6} + 2801218397375112 p^{75} T^{7} + p^{100} T^{8}
19C2S4C_2 \wr S_4 115008254302340928T+ 1 - 15008254302340928 T + 18 ⁣ ⁣0818\!\cdots\!08pT2 p T^{2} - 96 ⁣ ⁣9696\!\cdots\!96p2T3+ p^{2} T^{3} + 67 ⁣ ⁣6667\!\cdots\!66p3T4 p^{3} T^{4} - 96 ⁣ ⁣9696\!\cdots\!96p27T5+ p^{27} T^{5} + 18 ⁣ ⁣0818\!\cdots\!08p51T615008254302340928p75T7+p100T8 p^{51} T^{6} - 15008254302340928 p^{75} T^{7} + p^{100} T^{8}
23C2S4C_2 \wr S_4 1+48788561474090928T+ 1 + 48788561474090928 T + 10 ⁣ ⁣3610\!\cdots\!36pT2+ p T^{2} + 26 ⁣ ⁣8826\!\cdots\!88p2T3+ p^{2} T^{3} + 28 ⁣ ⁣3028\!\cdots\!30p3T4+ p^{3} T^{4} + 26 ⁣ ⁣8826\!\cdots\!88p27T5+ p^{27} T^{5} + 10 ⁣ ⁣3610\!\cdots\!36p51T6+48788561474090928p75T7+p100T8 p^{51} T^{6} + 48788561474090928 p^{75} T^{7} + p^{100} T^{8}
29C2S4C_2 \wr S_4 1366855400436538264T+ 1 - 366855400436538264 T + 37 ⁣ ⁣6037\!\cdots\!60T2 T^{2} - 81 ⁣ ⁣8881\!\cdots\!88T3+ T^{3} + 13 ⁣ ⁣7813\!\cdots\!78T4 T^{4} - 81 ⁣ ⁣8881\!\cdots\!88p25T5+ p^{25} T^{5} + 37 ⁣ ⁣6037\!\cdots\!60p50T6366855400436538264p75T7+p100T8 p^{50} T^{6} - 366855400436538264 p^{75} T^{7} + p^{100} T^{8}
31C2S4C_2 \wr S_4 15266475512689038864T+ 1 - 5266475512689038864 T + 25 ⁣ ⁣2825\!\cdots\!28T2 T^{2} - 36 ⁣ ⁣9236\!\cdots\!92T3+ T^{3} + 33 ⁣ ⁣3433\!\cdots\!34pT4 p T^{4} - 36 ⁣ ⁣9236\!\cdots\!92p25T5+ p^{25} T^{5} + 25 ⁣ ⁣2825\!\cdots\!28p50T65266475512689038864p75T7+p100T8 p^{50} T^{6} - 5266475512689038864 p^{75} T^{7} + p^{100} T^{8}
37C2S4C_2 \wr S_4 1+82582977571412504584T+ 1 + 82582977571412504584 T + 49 ⁣ ⁣5249\!\cdots\!52T2+ T^{2} + 21 ⁣ ⁣8421\!\cdots\!84T3+ T^{3} + 90 ⁣ ⁣3090\!\cdots\!30T4+ T^{4} + 21 ⁣ ⁣8421\!\cdots\!84p25T5+ p^{25} T^{5} + 49 ⁣ ⁣5249\!\cdots\!52p50T6+82582977571412504584p75T7+p100T8 p^{50} T^{6} + 82582977571412504584 p^{75} T^{7} + p^{100} T^{8}
41C2S4C_2 \wr S_4 1+ 1 + 11 ⁣ ⁣4411\!\cdots\!44T+ T + 68 ⁣ ⁣6868\!\cdots\!68T2+ T^{2} + 75 ⁣ ⁣9275\!\cdots\!92T3+ T^{3} + 20 ⁣ ⁣3420\!\cdots\!34T4+ T^{4} + 75 ⁣ ⁣9275\!\cdots\!92p25T5+ p^{25} T^{5} + 68 ⁣ ⁣6868\!\cdots\!68p50T6+ p^{50} T^{6} + 11 ⁣ ⁣4411\!\cdots\!44p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
43C2S4C_2 \wr S_4 1 1 - 29 ⁣ ⁣6029\!\cdots\!60T+ T + 24 ⁣ ⁣8024\!\cdots\!80T2 T^{2} - 55 ⁣ ⁣8055\!\cdots\!80T3+ T^{3} + 24 ⁣ ⁣9824\!\cdots\!98T4 T^{4} - 55 ⁣ ⁣8055\!\cdots\!80p25T5+ p^{25} T^{5} + 24 ⁣ ⁣8024\!\cdots\!80p50T6 p^{50} T^{6} - 29 ⁣ ⁣6029\!\cdots\!60p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
47C2S4C_2 \wr S_4 1 1 - 25 ⁣ ⁣2025\!\cdots\!20T+ T + 11 ⁣ ⁣2011\!\cdots\!20T2+ T^{2} + 22 ⁣ ⁣6022\!\cdots\!60T3+ T^{3} + 75 ⁣ ⁣9875\!\cdots\!98T4+ T^{4} + 22 ⁣ ⁣6022\!\cdots\!60p25T5+ p^{25} T^{5} + 11 ⁣ ⁣2011\!\cdots\!20p50T6 p^{50} T^{6} - 25 ⁣ ⁣2025\!\cdots\!20p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
53C2S4C_2 \wr S_4 1 1 - 17 ⁣ ⁣1217\!\cdots\!12T+ T + 37 ⁣ ⁣4837\!\cdots\!48T2 T^{2} - 50 ⁣ ⁣8850\!\cdots\!88T3+ T^{3} + 65 ⁣ ⁣3065\!\cdots\!30T4 T^{4} - 50 ⁣ ⁣8850\!\cdots\!88p25T5+ p^{25} T^{5} + 37 ⁣ ⁣4837\!\cdots\!48p50T6 p^{50} T^{6} - 17 ⁣ ⁣1217\!\cdots\!12p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
59C2S4C_2 \wr S_4 1+ 1 + 53 ⁣ ⁣3253\!\cdots\!32T+ T + 50 ⁣ ⁣9250\!\cdots\!92T2+ T^{2} + 20 ⁣ ⁣4420\!\cdots\!44T3+ T^{3} + 13 ⁣ ⁣1413\!\cdots\!14T4+ T^{4} + 20 ⁣ ⁣4420\!\cdots\!44p25T5+ p^{25} T^{5} + 50 ⁣ ⁣9250\!\cdots\!92p50T6+ p^{50} T^{6} + 53 ⁣ ⁣3253\!\cdots\!32p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
61C2S4C_2 \wr S_4 1 1 - 19 ⁣ ⁣0019\!\cdots\!00T+ T + 90 ⁣ ⁣7690\!\cdots\!76T2 T^{2} - 90 ⁣ ⁣0090\!\cdots\!00T3+ T^{3} + 36 ⁣ ⁣4636\!\cdots\!46T4 T^{4} - 90 ⁣ ⁣0090\!\cdots\!00p25T5+ p^{25} T^{5} + 90 ⁣ ⁣7690\!\cdots\!76p50T6 p^{50} T^{6} - 19 ⁣ ⁣0019\!\cdots\!00p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
67C2S4C_2 \wr S_4 1 1 - 22 ⁣ ⁣7222\!\cdots\!72T+ T + 28 ⁣ ⁣7228\!\cdots\!72T2 T^{2} - 24 ⁣ ⁣4024\!\cdots\!40T3+ T^{3} + 18 ⁣ ⁣8618\!\cdots\!86T4 T^{4} - 24 ⁣ ⁣4024\!\cdots\!40p25T5+ p^{25} T^{5} + 28 ⁣ ⁣7228\!\cdots\!72p50T6 p^{50} T^{6} - 22 ⁣ ⁣7222\!\cdots\!72p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
71C2S4C_2 \wr S_4 1+ 1 + 82 ⁣ ⁣0882\!\cdots\!08T+ T + 59 ⁣ ⁣2859\!\cdots\!28T2+ T^{2} + 37 ⁣ ⁣5637\!\cdots\!56T3+ T^{3} + 15 ⁣ ⁣7015\!\cdots\!70T4+ T^{4} + 37 ⁣ ⁣5637\!\cdots\!56p25T5+ p^{25} T^{5} + 59 ⁣ ⁣2859\!\cdots\!28p50T6+ p^{50} T^{6} + 82 ⁣ ⁣0882\!\cdots\!08p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
73C2S4C_2 \wr S_4 1 1 - 55 ⁣ ⁣6855\!\cdots\!68T+ T + 19 ⁣ ⁣2819\!\cdots\!28T2 T^{2} - 45 ⁣ ⁣7245\!\cdots\!72T3+ T^{3} + 94 ⁣ ⁣1094\!\cdots\!10T4 T^{4} - 45 ⁣ ⁣7245\!\cdots\!72p25T5+ p^{25} T^{5} + 19 ⁣ ⁣2819\!\cdots\!28p50T6 p^{50} T^{6} - 55 ⁣ ⁣6855\!\cdots\!68p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
79C2S4C_2 \wr S_4 1 1 - 11 ⁣ ⁣6011\!\cdots\!60T+ T + 13 ⁣ ⁣9613\!\cdots\!96T2 T^{2} - 89 ⁣ ⁣2089\!\cdots\!20T3+ T^{3} + 60 ⁣ ⁣0660\!\cdots\!06T4 T^{4} - 89 ⁣ ⁣2089\!\cdots\!20p25T5+ p^{25} T^{5} + 13 ⁣ ⁣9613\!\cdots\!96p50T6 p^{50} T^{6} - 11 ⁣ ⁣6011\!\cdots\!60p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
83C2S4C_2 \wr S_4 1+ 1 + 24 ⁣ ⁣0424\!\cdots\!04T+ T + 51 ⁣ ⁣8051\!\cdots\!80T2+ T^{2} + 68 ⁣ ⁣2468\!\cdots\!24T3+ T^{3} + 79 ⁣ ⁣8679\!\cdots\!86T4+ T^{4} + 68 ⁣ ⁣2468\!\cdots\!24p25T5+ p^{25} T^{5} + 51 ⁣ ⁣8051\!\cdots\!80p50T6+ p^{50} T^{6} + 24 ⁣ ⁣0424\!\cdots\!04p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
89C2S4C_2 \wr S_4 1+ 1 + 70 ⁣ ⁣0870\!\cdots\!08T+ T + 29 ⁣ ⁣9229\!\cdots\!92T2+ T^{2} + 99 ⁣ ⁣9699\!\cdots\!96T3+ T^{3} + 26 ⁣ ⁣1426\!\cdots\!14T4+ T^{4} + 99 ⁣ ⁣9699\!\cdots\!96p25T5+ p^{25} T^{5} + 29 ⁣ ⁣9229\!\cdots\!92p50T6+ p^{50} T^{6} + 70 ⁣ ⁣0870\!\cdots\!08p75T7+p100T8 p^{75} T^{7} + p^{100} T^{8}
97C2S4C_2 \wr S_4 1+ 1 + 15 ⁣ ⁣4815\!\cdots\!48T+ T + 25 ⁣ ⁣9225\!\cdots\!92T2+ T^{2} + 22 ⁣ ⁣2022\!\cdots\!20T3+ T^{3} + 19 ⁣ ⁣2619\!\cdots\!26T4+ T^{4} + 22 ⁣ ⁣2022\!\cdots\!20p25T5+ p^{25} T^{5} + 25 ⁣ ⁣9225\!\cdots\!92p50T6+ p^{50} T^{6} + 15 ⁣ ⁣4815\!\cdots\!48p75T7+p100T8 p^{75} T^{7} + p^{100} 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

−8.261778601273865218635055797278, −7.55763149504390845720637833992, −7.40242982521219263200210055907, −7.00798891434045335509220859301, −6.78349234785790289689252463220, −6.50432857365175385161024787479, −5.98317206398814351064296627023, −5.93073559791205231896715917495, −5.36407160517188913990754188299, −5.10883254847517096989700727407, −5.03932731376465296767391536262, −4.89466557570648250987038425641, −4.70116037586858864969836496570, −3.88407381251661718001439524430, −3.73450594082398097344142526678, −3.61906667808783979400479432925, −2.84110472395185403962082277847, −2.66676425648311612699477196126, −2.47671670759645185353059708988, −2.24755251728973214491999018172, −2.07097706028519140559650085514, −1.46856057943459527341415855780, −1.10866205417874722541528976439, −1.10520762670988228496773997418, −0.894192834311552323981584345390, 0, 0, 0, 0, 0.894192834311552323981584345390, 1.10520762670988228496773997418, 1.10866205417874722541528976439, 1.46856057943459527341415855780, 2.07097706028519140559650085514, 2.24755251728973214491999018172, 2.47671670759645185353059708988, 2.66676425648311612699477196126, 2.84110472395185403962082277847, 3.61906667808783979400479432925, 3.73450594082398097344142526678, 3.88407381251661718001439524430, 4.70116037586858864969836496570, 4.89466557570648250987038425641, 5.03932731376465296767391536262, 5.10883254847517096989700727407, 5.36407160517188913990754188299, 5.93073559791205231896715917495, 5.98317206398814351064296627023, 6.50432857365175385161024787479, 6.78349234785790289689252463220, 7.00798891434045335509220859301, 7.40242982521219263200210055907, 7.55763149504390845720637833992, 8.261778601273865218635055797278

Graph of the ZZ-function along the critical line