Properties

Label 8-588e4-1.1-c5e4-0-9
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

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 36·3-s + 810·9-s + 462·11-s + 602·13-s + 228·17-s + 358·19-s + 2.14e3·23-s − 3.52e3·25-s + 1.45e4·27-s − 5.53e3·29-s + 830·31-s + 1.66e4·33-s + 3.91e3·37-s + 2.16e4·39-s + 8.31e3·41-s − 1.45e4·43-s + 4.17e4·47-s + 8.20e3·51-s − 2.21e4·53-s + 1.28e4·57-s + 3.28e4·59-s + 8.37e4·61-s + 8.00e4·67-s + 7.73e4·69-s + 4.47e4·71-s − 2.24e4·73-s − 1.26e5·75-s + ⋯
L(s)  = 1  + 2.30·3-s + 10/3·9-s + 1.15·11-s + 0.987·13-s + 0.191·17-s + 0.227·19-s + 0.846·23-s − 1.12·25-s + 3.84·27-s − 1.22·29-s + 0.155·31-s + 2.65·33-s + 0.470·37-s + 2.28·39-s + 0.772·41-s − 1.19·43-s + 2.75·47-s + 0.441·51-s − 1.08·53-s + 0.525·57-s + 1.22·59-s + 2.88·61-s + 2.17·67-s + 1.95·69-s + 1.05·71-s − 0.493·73-s − 2.60·75-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 52.1995856552.19958565
L(12)L(\frac12) \approx 52.1995856552.19958565
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
3C1C_1 (1p2T)4 ( 1 - p^{2} T )^{4}
7 1 1
good5C2S4C_2 \wr S_4 1+3523T2132p4T3+13251176T4132p9T5+3523p10T6+p20T8 1 + 3523 T^{2} - 132 p^{4} T^{3} + 13251176 T^{4} - 132 p^{9} T^{5} + 3523 p^{10} T^{6} + p^{20} T^{8}
11C2S4C_2 \wr S_4 142pT+268705T250129610T3+18647530376T450129610p5T5+268705p10T642p16T7+p20T8 1 - 42 p T + 268705 T^{2} - 50129610 T^{3} + 18647530376 T^{4} - 50129610 p^{5} T^{5} + 268705 p^{10} T^{6} - 42 p^{16} T^{7} + p^{20} T^{8}
13C2S4C_2 \wr S_4 1602T12679pT2145984794T3+357680241104T4145984794p5T512679p11T6602p15T7+p20T8 1 - 602 T - 12679 p T^{2} - 145984794 T^{3} + 357680241104 T^{4} - 145984794 p^{5} T^{5} - 12679 p^{11} T^{6} - 602 p^{15} T^{7} + p^{20} T^{8}
17C2S4C_2 \wr S_4 1228T+3869284T21347363564T3+6940431119366T41347363564p5T5+3869284p10T6228p15T7+p20T8 1 - 228 T + 3869284 T^{2} - 1347363564 T^{3} + 6940431119366 T^{4} - 1347363564 p^{5} T^{5} + 3869284 p^{10} T^{6} - 228 p^{15} T^{7} + p^{20} T^{8}
19C2S4C_2 \wr S_4 1358T+7365277T22785970626T3+25078082046268T42785970626p5T5+7365277p10T6358p15T7+p20T8 1 - 358 T + 7365277 T^{2} - 2785970626 T^{3} + 25078082046268 T^{4} - 2785970626 p^{5} T^{5} + 7365277 p^{10} T^{6} - 358 p^{15} T^{7} + p^{20} T^{8}
23C2S4C_2 \wr S_4 12148T+16852828T216061095668T3+120569983154918T416061095668p5T5+16852828p10T62148p15T7+p20T8 1 - 2148 T + 16852828 T^{2} - 16061095668 T^{3} + 120569983154918 T^{4} - 16061095668 p^{5} T^{5} + 16852828 p^{10} T^{6} - 2148 p^{15} T^{7} + p^{20} T^{8}
29C2S4C_2 \wr S_4 1+5532T+56710687T2+131401968540T3+1151610033241988T4+131401968540p5T5+56710687p10T6+5532p15T7+p20T8 1 + 5532 T + 56710687 T^{2} + 131401968540 T^{3} + 1151610033241988 T^{4} + 131401968540 p^{5} T^{5} + 56710687 p^{10} T^{6} + 5532 p^{15} T^{7} + p^{20} T^{8}
31C2S4C_2 \wr S_4 1830T+31255592T2+217237417272T3+250607576745089T4+217237417272p5T5+31255592p10T6830p15T7+p20T8 1 - 830 T + 31255592 T^{2} + 217237417272 T^{3} + 250607576745089 T^{4} + 217237417272 p^{5} T^{5} + 31255592 p^{10} T^{6} - 830 p^{15} T^{7} + p^{20} T^{8}
37C2S4C_2 \wr S_4 13914T+81854461T2709188215618T3+5749597008366352T4709188215618p5T5+81854461p10T63914p15T7+p20T8 1 - 3914 T + 81854461 T^{2} - 709188215618 T^{3} + 5749597008366352 T^{4} - 709188215618 p^{5} T^{5} + 81854461 p^{10} T^{6} - 3914 p^{15} T^{7} + p^{20} T^{8}
41C2S4C_2 \wr S_4 18316T+131719112T2+1793338464780T311507939827736466T4+1793338464780p5T5+131719112p10T68316p15T7+p20T8 1 - 8316 T + 131719112 T^{2} + 1793338464780 T^{3} - 11507939827736466 T^{4} + 1793338464780 p^{5} T^{5} + 131719112 p^{10} T^{6} - 8316 p^{15} T^{7} + p^{20} T^{8}
43C2S4C_2 \wr S_4 1+14518T+553805833T2+5594375858722T3+119245786138973608T4+5594375858722p5T5+553805833p10T6+14518p15T7+p20T8 1 + 14518 T + 553805833 T^{2} + 5594375858722 T^{3} + 119245786138973608 T^{4} + 5594375858722 p^{5} T^{5} + 553805833 p^{10} T^{6} + 14518 p^{15} T^{7} + p^{20} T^{8}
47C2S4C_2 \wr S_4 141700T+1201404476T2465323341500pT3+369215501257269942T4465323341500p6T5+1201404476p10T641700p15T7+p20T8 1 - 41700 T + 1201404476 T^{2} - 465323341500 p T^{3} + 369215501257269942 T^{4} - 465323341500 p^{6} T^{5} + 1201404476 p^{10} T^{6} - 41700 p^{15} T^{7} + p^{20} T^{8}
53C2S4C_2 \wr S_4 1+22164T+612500711T29154492772292T3135301388975352444T49154492772292p5T5+612500711p10T6+22164p15T7+p20T8 1 + 22164 T + 612500711 T^{2} - 9154492772292 T^{3} - 135301388975352444 T^{4} - 9154492772292 p^{5} T^{5} + 612500711 p^{10} T^{6} + 22164 p^{15} T^{7} + p^{20} T^{8}
59C2S4C_2 \wr S_4 132886T+2165272693T253279788989366T3+2201676784894672940T453279788989366p5T5+2165272693p10T632886p15T7+p20T8 1 - 32886 T + 2165272693 T^{2} - 53279788989366 T^{3} + 2201676784894672940 T^{4} - 53279788989366 p^{5} T^{5} + 2165272693 p^{10} T^{6} - 32886 p^{15} T^{7} + p^{20} T^{8}
61C2S4C_2 \wr S_4 183732T+5138847044T2213873087755964T3+7191474746319250502T4213873087755964p5T5+5138847044p10T683732p15T7+p20T8 1 - 83732 T + 5138847044 T^{2} - 213873087755964 T^{3} + 7191474746319250502 T^{4} - 213873087755964 p^{5} T^{5} + 5138847044 p^{10} T^{6} - 83732 p^{15} T^{7} + p^{20} T^{8}
67C2S4C_2 \wr S_4 180034T+6110296361T2278312842674138T3+12284801359836541152T4278312842674138p5T5+6110296361p10T680034p15T7+p20T8 1 - 80034 T + 6110296361 T^{2} - 278312842674138 T^{3} + 12284801359836541152 T^{4} - 278312842674138 p^{5} T^{5} + 6110296361 p^{10} T^{6} - 80034 p^{15} T^{7} + p^{20} T^{8}
71C2S4C_2 \wr S_4 144772T+5603702956T2219545846683812T3+14130153190557486902T4219545846683812p5T5+5603702956p10T644772p15T7+p20T8 1 - 44772 T + 5603702956 T^{2} - 219545846683812 T^{3} + 14130153190557486902 T^{4} - 219545846683812 p^{5} T^{5} + 5603702956 p^{10} T^{6} - 44772 p^{15} T^{7} + p^{20} T^{8}
73C2S4C_2 \wr S_4 1+22470T+3732785609T2+14345610615750T3+8330122733588006676T4+14345610615750p5T5+3732785609p10T6+22470p15T7+p20T8 1 + 22470 T + 3732785609 T^{2} + 14345610615750 T^{3} + 8330122733588006676 T^{4} + 14345610615750 p^{5} T^{5} + 3732785609 p^{10} T^{6} + 22470 p^{15} T^{7} + p^{20} T^{8}
79C2S4C_2 \wr S_4 175286T+10257886376T2522030297673368T3+42333862433009559377T4522030297673368p5T5+10257886376p10T675286p15T7+p20T8 1 - 75286 T + 10257886376 T^{2} - 522030297673368 T^{3} + 42333862433009559377 T^{4} - 522030297673368 p^{5} T^{5} + 10257886376 p^{10} T^{6} - 75286 p^{15} T^{7} + p^{20} T^{8}
83C2S4C_2 \wr S_4 117418T+11662792993T2252243184391142T3+61381737302005449608T4252243184391142p5T5+11662792993p10T617418p15T7+p20T8 1 - 17418 T + 11662792993 T^{2} - 252243184391142 T^{3} + 61381737302005449608 T^{4} - 252243184391142 p^{5} T^{5} + 11662792993 p^{10} T^{6} - 17418 p^{15} T^{7} + p^{20} T^{8}
89C2S4C_2 \wr S_4 128944T+4985253824T2135699519340656T36573734910358628322T4135699519340656p5T5+4985253824p10T628944p15T7+p20T8 1 - 28944 T + 4985253824 T^{2} - 135699519340656 T^{3} - 6573734910358628322 T^{4} - 135699519340656 p^{5} T^{5} + 4985253824 p^{10} T^{6} - 28944 p^{15} T^{7} + p^{20} T^{8}
97C2S4C_2 \wr S_4 1216678T+43602999305T24974916579453302T3+ 1 - 216678 T + 43602999305 T^{2} - 4974916579453302 T^{3} + 56 ⁣ ⁣0056\!\cdots\!00T44974916579453302p5T5+43602999305p10T6216678p15T7+p20T8 T^{4} - 4974916579453302 p^{5} T^{5} + 43602999305 p^{10} T^{6} - 216678 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

−7.18102493425257521845906584047, −6.69808121762116658713841796084, −6.36687338852324465801638260491, −6.36598688716242558748899118855, −6.07676048605494948911704554864, −5.65305201763409090172770186779, −5.28771184342674708876054141718, −5.24428039898948773581046069210, −5.03250606626888387137054384718, −4.25695826403004166418053211987, −4.16156170581148714606795259175, −4.07052994801156078545224181445, −4.06828331719991921553191082911, −3.42814849294869845924327700258, −3.28893880922404834078714316081, −3.07503857938357793721555985406, −3.04909450570473859424203583454, −2.11586744594484482600895167391, −2.09448299306356188416019700470, −2.05406851904030609848654173438, −1.89835545859806929760184414904, −1.14794996546507994633999169744, −0.879963474089920637508027276258, −0.64547434166418236436075483797, −0.62230237170248182907945090259, 0.62230237170248182907945090259, 0.64547434166418236436075483797, 0.879963474089920637508027276258, 1.14794996546507994633999169744, 1.89835545859806929760184414904, 2.05406851904030609848654173438, 2.09448299306356188416019700470, 2.11586744594484482600895167391, 3.04909450570473859424203583454, 3.07503857938357793721555985406, 3.28893880922404834078714316081, 3.42814849294869845924327700258, 4.06828331719991921553191082911, 4.07052994801156078545224181445, 4.16156170581148714606795259175, 4.25695826403004166418053211987, 5.03250606626888387137054384718, 5.24428039898948773581046069210, 5.28771184342674708876054141718, 5.65305201763409090172770186779, 6.07676048605494948911704554864, 6.36598688716242558748899118855, 6.36687338852324465801638260491, 6.69808121762116658713841796084, 7.18102493425257521845906584047

Graph of the ZZ-function along the critical line