Properties

Label 8-588e4-1.1-c5e4-0-1
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

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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 0.046411012620.04641101262
L(12)L(\frac12) \approx 0.046411012620.04641101262
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 (1+p2T)4 ( 1 + p^{2} T )^{4}
7 1 1
good5C2S4C_2 \wr S_4 1+3523T2+132p4T3+13251176T4+132p9T5+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 1+602T12679pT2+145984794T3+357680241104T4+145984794p5T512679p11T6+602p15T7+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 1+228T+3869284T2+1347363564T3+6940431119366T4+1347363564p5T5+3869284p10T6+228p15T7+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 1+358T+7365277T2+2785970626T3+25078082046268T4+2785970626p5T5+7365277p10T6+358p15T7+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 1+830T+31255592T2217237417272T3+250607576745089T4217237417272p5T5+31255592p10T6+830p15T7+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 1+8316T+131719112T21793338464780T311507939827736466T41793338464780p5T5+131719112p10T6+8316p15T7+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 1+41700T+1201404476T2+465323341500pT3+369215501257269942T4+465323341500p6T5+1201404476p10T6+41700p15T7+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 1+32886T+2165272693T2+53279788989366T3+2201676784894672940T4+53279788989366p5T5+2165272693p10T6+32886p15T7+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 1+83732T+5138847044T2+213873087755964T3+7191474746319250502T4+213873087755964p5T5+5138847044p10T6+83732p15T7+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 122470T+3732785609T214345610615750T3+8330122733588006676T414345610615750p5T5+3732785609p10T622470p15T7+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 1+17418T+11662792993T2+252243184391142T3+61381737302005449608T4+252243184391142p5T5+11662792993p10T6+17418p15T7+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 1+28944T+4985253824T2+135699519340656T36573734910358628322T4+135699519340656p5T5+4985253824p10T6+28944p15T7+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 1+216678T+43602999305T2+4974916579453302T3+ 1 + 216678 T + 43602999305 T^{2} + 4974916579453302 T^{3} + 56 ⁣ ⁣0056\!\cdots\!00T4+4974916579453302p5T5+43602999305p10T6+216678p15T7+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

−6.80631305418479735947714268971, −6.44844864753482093055307576236, −6.42091176549015319922998194470, −6.26383618306861954487667524131, −6.09756040032114426162042649551, −5.57831951285068741473951625650, −5.32469718375663487150801739064, −5.30618578983063020268865576510, −5.10220746236652971520742976048, −4.68324149007996098994271096569, −4.38271679315724081885495903192, −4.32774520428248031298180763401, −4.23858047685534849350317826326, −3.44234435551543409629463844544, −3.43523010573100753682072297173, −3.20610846984594359584882355082, −2.93010157898980910543318897937, −2.07640048635663732507483068213, −1.91963735283839528513543943716, −1.78907495028052741552925999890, −1.61516258369381417752183062074, −1.05591681552149206355491521692, −0.60987739235975743285295035149, −0.56103016638208437468744430182, −0.04074361905608839665991496910, 0.04074361905608839665991496910, 0.56103016638208437468744430182, 0.60987739235975743285295035149, 1.05591681552149206355491521692, 1.61516258369381417752183062074, 1.78907495028052741552925999890, 1.91963735283839528513543943716, 2.07640048635663732507483068213, 2.93010157898980910543318897937, 3.20610846984594359584882355082, 3.43523010573100753682072297173, 3.44234435551543409629463844544, 4.23858047685534849350317826326, 4.32774520428248031298180763401, 4.38271679315724081885495903192, 4.68324149007996098994271096569, 5.10220746236652971520742976048, 5.30618578983063020268865576510, 5.32469718375663487150801739064, 5.57831951285068741473951625650, 6.09756040032114426162042649551, 6.26383618306861954487667524131, 6.42091176549015319922998194470, 6.44844864753482093055307576236, 6.80631305418479735947714268971

Graph of the ZZ-function along the critical line