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Properties

Label 8-693e4-1.1-c3e4-0-1
Degree 88
Conductor 230639102001230639102001
Sign 11
Analytic cond. 2.79509×1062.79509\times 10^{6}
Root an. cond. 6.394396.39439
Motivic weight 33
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  + 4·2-s + 3·4-s + 18·5-s − 28·7-s + 72·10-s − 44·11-s − 134·13-s − 112·14-s − 25·16-s + 74·17-s − 164·19-s + 54·20-s − 176·22-s − 194·23-s − 69·25-s − 536·26-s − 84·28-s + 108·29-s − 412·31-s − 244·32-s + 296·34-s − 504·35-s + 286·37-s − 656·38-s + 18·41-s − 496·43-s − 132·44-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/8·4-s + 1.60·5-s − 1.51·7-s + 2.27·10-s − 1.20·11-s − 2.85·13-s − 2.13·14-s − 0.390·16-s + 1.05·17-s − 1.98·19-s + 0.603·20-s − 1.70·22-s − 1.75·23-s − 0.551·25-s − 4.04·26-s − 0.566·28-s + 0.691·29-s − 2.38·31-s − 1.34·32-s + 1.49·34-s − 2.43·35-s + 1.27·37-s − 2.80·38-s + 0.0685·41-s − 1.75·43-s − 0.452·44-s + ⋯

Functional equation

Λ(s)=((38⋅74⋅114)s/2 ΓC(s)4 L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
Λ(s)=((38⋅74⋅114)s/2 ΓC(s+3/2)4 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 88
Conductor: 38⋅74⋅1143^{8} \cdot 7^{4} \cdot 11^{4}
Sign: 11
Analytic conductor: 2.79509×1062.79509\times 10^{6}
Root analytic conductor: 6.394396.39439
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 44
Selberg data: (8, 38⋅74⋅114, ( :3/2,3/2,3/2,3/2), 1)(8,\ 3^{8} \cdot 7^{4} \cdot 11^{4} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{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)
bad3 1 1
7C1C_1 (1+pT)4 ( 1 + p T )^{4}
11C1C_1 (1+pT)4 ( 1 + p T )^{4}
good2C2≀S4C_2 \wr S_4 1−p2T+13T2−5p3T3+73pT4−5p6T5+13p6T6−p11T7+p12T8 1 - p^{2} T + 13 T^{2} - 5 p^{3} T^{3} + 73 p T^{4} - 5 p^{6} T^{5} + 13 p^{6} T^{6} - p^{11} T^{7} + p^{12} T^{8}
5C2≀S4C_2 \wr S_4 1−18T+393T2−758pT3+56148T4−758p4T5+393p6T6−18p9T7+p12T8 1 - 18 T + 393 T^{2} - 758 p T^{3} + 56148 T^{4} - 758 p^{4} T^{5} + 393 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8}
13C2≀S4C_2 \wr S_4 1+134T+10520T2+559034T3+26667070T4+559034p3T5+10520p6T6+134p9T7+p12T8 1 + 134 T + 10520 T^{2} + 559034 T^{3} + 26667070 T^{4} + 559034 p^{3} T^{5} + 10520 p^{6} T^{6} + 134 p^{9} T^{7} + p^{12} T^{8}
17C2≀S4C_2 \wr S_4 1−74T+17468T2−831750T3+118657126T4−831750p3T5+17468p6T6−74p9T7+p12T8 1 - 74 T + 17468 T^{2} - 831750 T^{3} + 118657126 T^{4} - 831750 p^{3} T^{5} + 17468 p^{6} T^{6} - 74 p^{9} T^{7} + p^{12} T^{8}
19C2≀S4C_2 \wr S_4 1+164T+11968T2+138820T3−15467058T4+138820p3T5+11968p6T6+164p9T7+p12T8 1 + 164 T + 11968 T^{2} + 138820 T^{3} - 15467058 T^{4} + 138820 p^{3} T^{5} + 11968 p^{6} T^{6} + 164 p^{9} T^{7} + p^{12} T^{8}
23C2≀S4C_2 \wr S_4 1+194T+46053T2+5374478T3+784861428T4+5374478p3T5+46053p6T6+194p9T7+p12T8 1 + 194 T + 46053 T^{2} + 5374478 T^{3} + 784861428 T^{4} + 5374478 p^{3} T^{5} + 46053 p^{6} T^{6} + 194 p^{9} T^{7} + p^{12} T^{8}
29C2≀S4C_2 \wr S_4 1−108T+42724T2+1514908T3+528463590T4+1514908p3T5+42724p6T6−108p9T7+p12T8 1 - 108 T + 42724 T^{2} + 1514908 T^{3} + 528463590 T^{4} + 1514908 p^{3} T^{5} + 42724 p^{6} T^{6} - 108 p^{9} T^{7} + p^{12} T^{8}
31C2≀S4C_2 \wr S_4 1+412T+51613T2−11046366T3−3907770320T4−11046366p3T5+51613p6T6+412p9T7+p12T8 1 + 412 T + 51613 T^{2} - 11046366 T^{3} - 3907770320 T^{4} - 11046366 p^{3} T^{5} + 51613 p^{6} T^{6} + 412 p^{9} T^{7} + p^{12} T^{8}
37C2≀S4C_2 \wr S_4 1−286T+135513T2−31935490T3+9691467156T4−31935490p3T5+135513p6T6−286p9T7+p12T8 1 - 286 T + 135513 T^{2} - 31935490 T^{3} + 9691467156 T^{4} - 31935490 p^{3} T^{5} + 135513 p^{6} T^{6} - 286 p^{9} T^{7} + p^{12} T^{8}
41C2≀S4C_2 \wr S_4 1−18T+114524T2+18203666T3+5875037446T4+18203666p3T5+114524p6T6−18p9T7+p12T8 1 - 18 T + 114524 T^{2} + 18203666 T^{3} + 5875037446 T^{4} + 18203666 p^{3} T^{5} + 114524 p^{6} T^{6} - 18 p^{9} T^{7} + p^{12} T^{8}
43C2≀S4C_2 \wr S_4 1+496T+308268T2+103155312T3+35378135478T4+103155312p3T5+308268p6T6+496p9T7+p12T8 1 + 496 T + 308268 T^{2} + 103155312 T^{3} + 35378135478 T^{4} + 103155312 p^{3} T^{5} + 308268 p^{6} T^{6} + 496 p^{9} T^{7} + p^{12} T^{8}
47C2≀S4C_2 \wr S_4 1+62T+345832T2+21522086T3+50715680878T4+21522086p3T5+345832p6T6+62p9T7+p12T8 1 + 62 T + 345832 T^{2} + 21522086 T^{3} + 50715680878 T^{4} + 21522086 p^{3} T^{5} + 345832 p^{6} T^{6} + 62 p^{9} T^{7} + p^{12} T^{8}
53C2≀S4C_2 \wr S_4 1−828T+705892T2−360136820T3+165458948486T4−360136820p3T5+705892p6T6−828p9T7+p12T8 1 - 828 T + 705892 T^{2} - 360136820 T^{3} + 165458948486 T^{4} - 360136820 p^{3} T^{5} + 705892 p^{6} T^{6} - 828 p^{9} T^{7} + p^{12} T^{8}
59C2≀S4C_2 \wr S_4 1−1224T+1275413T2−804281630T3+441199752912T4−804281630p3T5+1275413p6T6−1224p9T7+p12T8 1 - 1224 T + 1275413 T^{2} - 804281630 T^{3} + 441199752912 T^{4} - 804281630 p^{3} T^{5} + 1275413 p^{6} T^{6} - 1224 p^{9} T^{7} + p^{12} T^{8}
61C2≀S4C_2 \wr S_4 1+350T+430588T2+138974618T3+88699741846T4+138974618p3T5+430588p6T6+350p9T7+p12T8 1 + 350 T + 430588 T^{2} + 138974618 T^{3} + 88699741846 T^{4} + 138974618 p^{3} T^{5} + 430588 p^{6} T^{6} + 350 p^{9} T^{7} + p^{12} T^{8}
67C2≀S4C_2 \wr S_4 1+1498T+1714917T2+1237137334T3+793168291468T4+1237137334p3T5+1714917p6T6+1498p9T7+p12T8 1 + 1498 T + 1714917 T^{2} + 1237137334 T^{3} + 793168291468 T^{4} + 1237137334 p^{3} T^{5} + 1714917 p^{6} T^{6} + 1498 p^{9} T^{7} + p^{12} T^{8}
71C2≀S4C_2 \wr S_4 1+2326T+3445789T2+3267789594T3+2320231641060T4+3267789594p3T5+3445789p6T6+2326p9T7+p12T8 1 + 2326 T + 3445789 T^{2} + 3267789594 T^{3} + 2320231641060 T^{4} + 3267789594 p^{3} T^{5} + 3445789 p^{6} T^{6} + 2326 p^{9} T^{7} + p^{12} T^{8}
73C2≀S4C_2 \wr S_4 1+1630T+2286292T2+1913055042T3+1441611662422T4+1913055042p3T5+2286292p6T6+1630p9T7+p12T8 1 + 1630 T + 2286292 T^{2} + 1913055042 T^{3} + 1441611662422 T^{4} + 1913055042 p^{3} T^{5} + 2286292 p^{6} T^{6} + 1630 p^{9} T^{7} + p^{12} T^{8}
79C2≀S4C_2 \wr S_4 1+1020T+1989936T2+1295128300T3+1427479601566T4+1295128300p3T5+1989936p6T6+1020p9T7+p12T8 1 + 1020 T + 1989936 T^{2} + 1295128300 T^{3} + 1427479601566 T^{4} + 1295128300 p^{3} T^{5} + 1989936 p^{6} T^{6} + 1020 p^{9} T^{7} + p^{12} T^{8}
83C2≀S4C_2 \wr S_4 1−1920T+2662892T2−2752584832T3+2433754978902T4−2752584832p3T5+2662892p6T6−1920p9T7+p12T8 1 - 1920 T + 2662892 T^{2} - 2752584832 T^{3} + 2433754978902 T^{4} - 2752584832 p^{3} T^{5} + 2662892 p^{6} T^{6} - 1920 p^{9} T^{7} + p^{12} T^{8}
89C2≀S4C_2 \wr S_4 1+1550T+1862281T2+874387910T3+705085008356T4+874387910p3T5+1862281p6T6+1550p9T7+p12T8 1 + 1550 T + 1862281 T^{2} + 874387910 T^{3} + 705085008356 T^{4} + 874387910 p^{3} T^{5} + 1862281 p^{6} T^{6} + 1550 p^{9} T^{7} + p^{12} T^{8}
97C2≀S4C_2 \wr S_4 1+2202T+5018625T2+6116849198T3+7416588769820T4+6116849198p3T5+5018625p6T6+2202p9T7+p12T8 1 + 2202 T + 5018625 T^{2} + 6116849198 T^{3} + 7416588769820 T^{4} + 6116849198 p^{3} T^{5} + 5018625 p^{6} T^{6} + 2202 p^{9} T^{7} + p^{12} 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

−7.43908560853462589685585361744, −7.42069251850984945151730663946, −7.02836232436908219879105323332, −6.88384612999257518466981169394, −6.48999357559035646455439210864, −6.25607465360071010486670127776, −5.97908099531001712952082097493, −5.83209613863528777573237939348, −5.72566311713299537455326859646, −5.37709473805517550820976300062, −5.27849320457487709593338216840, −4.96037070827037694424298004050, −4.77636070803062740776977182768, −4.26722120098286793240652316473, −4.17270610490476072193330805663, −4.01275152710589195220119942013, −3.89631367850701075533333192742, −3.20097051193696083071281728439, −2.89011072785254005961638972133, −2.87829508660891421682969294003, −2.27790683984819339740675165412, −2.23792866836422627527643145512, −2.16301159074055280804543063375, −1.48526244251488619030461131691, −1.34468543103861334978274148012, 0, 0, 0, 0, 1.34468543103861334978274148012, 1.48526244251488619030461131691, 2.16301159074055280804543063375, 2.23792866836422627527643145512, 2.27790683984819339740675165412, 2.87829508660891421682969294003, 2.89011072785254005961638972133, 3.20097051193696083071281728439, 3.89631367850701075533333192742, 4.01275152710589195220119942013, 4.17270610490476072193330805663, 4.26722120098286793240652316473, 4.77636070803062740776977182768, 4.96037070827037694424298004050, 5.27849320457487709593338216840, 5.37709473805517550820976300062, 5.72566311713299537455326859646, 5.83209613863528777573237939348, 5.97908099531001712952082097493, 6.25607465360071010486670127776, 6.48999357559035646455439210864, 6.88384612999257518466981169394, 7.02836232436908219879105323332, 7.42069251850984945151730663946, 7.43908560853462589685585361744

Graph of the ZZ-function along the critical line