Properties

Label 16-14e8-1.1-c13e8-0-1
Degree $16$
Conductor $1475789056$
Sign $1$
Analytic cond. $2.57979\times 10^{9}$
Root an. cond. $3.87457$
Motivic weight $13$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 256·2-s − 182·3-s + 2.45e4·4-s − 6.44e4·5-s + 4.65e4·6-s − 1.13e5·7-s + 1.21e6·9-s + 1.64e7·10-s − 1.00e6·11-s − 4.47e6·12-s + 5.38e7·13-s + 2.91e7·14-s + 1.17e7·15-s − 2.51e8·16-s − 1.65e8·17-s − 3.10e8·18-s − 4.23e8·19-s − 1.58e9·20-s + 2.06e7·21-s + 2.58e8·22-s + 2.86e8·23-s + 4.27e9·25-s − 1.37e10·26-s − 3.88e9·27-s − 2.79e9·28-s + 1.82e10·29-s − 3.00e9·30-s + ⋯
L(s)  = 1  − 2.82·2-s − 0.144·3-s + 3·4-s − 1.84·5-s + 0.407·6-s − 0.365·7-s + 0.761·9-s + 5.21·10-s − 0.171·11-s − 0.432·12-s + 3.09·13-s + 1.03·14-s + 0.265·15-s − 3.75·16-s − 1.66·17-s − 2.15·18-s − 2.06·19-s − 5.52·20-s + 0.0526·21-s + 0.485·22-s + 0.403·23-s + 3.50·25-s − 8.74·26-s − 1.92·27-s − 1.09·28-s + 5.68·29-s − 0.751·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(14-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+13/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.57979\times 10^{9}\)
Root analytic conductor: \(3.87457\)
Motivic weight: \(13\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{14} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [13/2]^{8} ),\ 1 )\)

Particular Values

\(L(7)\) \(\approx\) \(0.6854418617\)
\(L(\frac12)\) \(\approx\) \(0.6854418617\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 + p^{6} T + p^{12} T^{2} )^{4} \)
7 \( 1 + 16248 p T + 107711924 p^{3} T^{2} - 5716640280 p^{8} T^{3} - 20909219190 p^{13} T^{4} - 5716640280 p^{21} T^{5} + 107711924 p^{29} T^{6} + 16248 p^{40} T^{7} + p^{52} T^{8} \)
good3 \( 1 + 182 T - 1180208 T^{2} + 1148428120 p T^{3} - 25485034301 p^{2} T^{4} - 64848767574836 p^{4} T^{5} + 9348143118724660 p^{6} T^{6} + 219482061877171478 p^{9} T^{7} - 19002836451041603312 p^{12} T^{8} + 219482061877171478 p^{22} T^{9} + 9348143118724660 p^{32} T^{10} - 64848767574836 p^{43} T^{11} - 25485034301 p^{54} T^{12} + 1148428120 p^{66} T^{13} - 1180208 p^{78} T^{14} + 182 p^{91} T^{15} + p^{104} T^{16} \)
5 \( 1 + 2576 p^{2} T - 131717534 T^{2} - 18469552348464 p T^{3} - 1271390721497581219 T^{4} + \)\(30\!\cdots\!28\)\( p T^{5} - \)\(38\!\cdots\!26\)\( p^{2} T^{6} + \)\(46\!\cdots\!84\)\( p^{4} T^{7} + \)\(26\!\cdots\!04\)\( p^{6} T^{8} + \)\(46\!\cdots\!84\)\( p^{17} T^{9} - \)\(38\!\cdots\!26\)\( p^{28} T^{10} + \)\(30\!\cdots\!28\)\( p^{40} T^{11} - 1271390721497581219 p^{52} T^{12} - 18469552348464 p^{66} T^{13} - 131717534 p^{78} T^{14} + 2576 p^{93} T^{15} + p^{104} T^{16} \)
11 \( 1 + 1008790 T - 99056291003072 T^{2} - 25923801408883395432 p T^{3} + \)\(36\!\cdots\!67\)\( p^{4} T^{4} + \)\(12\!\cdots\!48\)\( p^{3} T^{5} - \)\(12\!\cdots\!24\)\( p^{4} T^{6} - \)\(19\!\cdots\!02\)\( p^{5} T^{7} + \)\(30\!\cdots\!48\)\( p^{6} T^{8} - \)\(19\!\cdots\!02\)\( p^{18} T^{9} - \)\(12\!\cdots\!24\)\( p^{30} T^{10} + \)\(12\!\cdots\!48\)\( p^{42} T^{11} + \)\(36\!\cdots\!67\)\( p^{56} T^{12} - 25923801408883395432 p^{66} T^{13} - 99056291003072 p^{78} T^{14} + 1008790 p^{91} T^{15} + p^{104} T^{16} \)
13 \( ( 1 - 26903632 T + 97747020241532 p T^{2} - \)\(22\!\cdots\!96\)\( T^{3} + \)\(57\!\cdots\!98\)\( T^{4} - \)\(22\!\cdots\!96\)\( p^{13} T^{5} + 97747020241532 p^{27} T^{6} - 26903632 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
17 \( 1 + 165333028 T + 14301100946612206 T^{2} - \)\(30\!\cdots\!72\)\( p T^{3} - \)\(14\!\cdots\!51\)\( p T^{4} - \)\(33\!\cdots\!24\)\( T^{5} - \)\(37\!\cdots\!06\)\( T^{6} + \)\(28\!\cdots\!48\)\( T^{7} + \)\(46\!\cdots\!76\)\( T^{8} + \)\(28\!\cdots\!48\)\( p^{13} T^{9} - \)\(37\!\cdots\!06\)\( p^{26} T^{10} - \)\(33\!\cdots\!24\)\( p^{39} T^{11} - \)\(14\!\cdots\!51\)\( p^{53} T^{12} - \)\(30\!\cdots\!72\)\( p^{66} T^{13} + 14301100946612206 p^{78} T^{14} + 165333028 p^{91} T^{15} + p^{104} T^{16} \)
19 \( 1 + 423405794 T + 5480059521660384 T^{2} - \)\(11\!\cdots\!36\)\( T^{3} + \)\(15\!\cdots\!67\)\( T^{4} + \)\(83\!\cdots\!40\)\( T^{5} - \)\(20\!\cdots\!52\)\( T^{6} - \)\(39\!\cdots\!70\)\( T^{7} + \)\(92\!\cdots\!88\)\( T^{8} - \)\(39\!\cdots\!70\)\( p^{13} T^{9} - \)\(20\!\cdots\!52\)\( p^{26} T^{10} + \)\(83\!\cdots\!40\)\( p^{39} T^{11} + \)\(15\!\cdots\!67\)\( p^{52} T^{12} - \)\(11\!\cdots\!36\)\( p^{65} T^{13} + 5480059521660384 p^{78} T^{14} + 423405794 p^{91} T^{15} + p^{104} T^{16} \)
23 \( 1 - 286233866 T - 881189351639823284 T^{2} - \)\(24\!\cdots\!64\)\( T^{3} + \)\(41\!\cdots\!39\)\( T^{4} + \)\(26\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!80\)\( T^{6} - \)\(10\!\cdots\!46\)\( T^{7} - \)\(24\!\cdots\!20\)\( T^{8} - \)\(10\!\cdots\!46\)\( p^{13} T^{9} + \)\(11\!\cdots\!80\)\( p^{26} T^{10} + \)\(26\!\cdots\!72\)\( p^{39} T^{11} + \)\(41\!\cdots\!39\)\( p^{52} T^{12} - \)\(24\!\cdots\!64\)\( p^{65} T^{13} - 881189351639823284 p^{78} T^{14} - 286233866 p^{91} T^{15} + p^{104} T^{16} \)
29 \( ( 1 - 9100337408 T + 63557643354325261932 T^{2} - \)\(29\!\cdots\!08\)\( T^{3} + \)\(10\!\cdots\!82\)\( T^{4} - \)\(29\!\cdots\!08\)\( p^{13} T^{5} + 63557643354325261932 p^{26} T^{6} - 9100337408 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
31 \( 1 - 1507094246 T - 84905048460002200900 T^{2} + \)\(56\!\cdots\!56\)\( T^{3} + \)\(43\!\cdots\!95\)\( T^{4} - \)\(12\!\cdots\!32\)\( T^{5} - \)\(15\!\cdots\!68\)\( T^{6} + \)\(12\!\cdots\!50\)\( T^{7} + \)\(43\!\cdots\!44\)\( T^{8} + \)\(12\!\cdots\!50\)\( p^{13} T^{9} - \)\(15\!\cdots\!68\)\( p^{26} T^{10} - \)\(12\!\cdots\!32\)\( p^{39} T^{11} + \)\(43\!\cdots\!95\)\( p^{52} T^{12} + \)\(56\!\cdots\!56\)\( p^{65} T^{13} - 84905048460002200900 p^{78} T^{14} - 1507094246 p^{91} T^{15} + p^{104} T^{16} \)
37 \( 1 + 18959705336 T - \)\(38\!\cdots\!50\)\( T^{2} - \)\(20\!\cdots\!04\)\( T^{3} + \)\(17\!\cdots\!49\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{5} - \)\(47\!\cdots\!34\)\( T^{6} - \)\(34\!\cdots\!28\)\( T^{7} + \)\(65\!\cdots\!60\)\( T^{8} - \)\(34\!\cdots\!28\)\( p^{13} T^{9} - \)\(47\!\cdots\!34\)\( p^{26} T^{10} - \)\(38\!\cdots\!24\)\( p^{39} T^{11} + \)\(17\!\cdots\!49\)\( p^{52} T^{12} - \)\(20\!\cdots\!04\)\( p^{65} T^{13} - \)\(38\!\cdots\!50\)\( p^{78} T^{14} + 18959705336 p^{91} T^{15} + p^{104} T^{16} \)
41 \( ( 1 - 56825955312 T + \)\(21\!\cdots\!32\)\( T^{2} - \)\(40\!\cdots\!08\)\( T^{3} + \)\(12\!\cdots\!54\)\( T^{4} - \)\(40\!\cdots\!08\)\( p^{13} T^{5} + \)\(21\!\cdots\!32\)\( p^{26} T^{6} - 56825955312 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
43 \( ( 1 + 45778809712 T + \)\(48\!\cdots\!88\)\( T^{2} + \)\(17\!\cdots\!40\)\( T^{3} + \)\(10\!\cdots\!70\)\( T^{4} + \)\(17\!\cdots\!40\)\( p^{13} T^{5} + \)\(48\!\cdots\!88\)\( p^{26} T^{6} + 45778809712 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
47 \( 1 + 81351201078 T - \)\(58\!\cdots\!84\)\( T^{2} + \)\(59\!\cdots\!56\)\( T^{3} + \)\(90\!\cdots\!83\)\( T^{4} - \)\(40\!\cdots\!84\)\( T^{5} + \)\(97\!\cdots\!84\)\( T^{6} + \)\(24\!\cdots\!18\)\( T^{7} - \)\(13\!\cdots\!84\)\( T^{8} + \)\(24\!\cdots\!18\)\( p^{13} T^{9} + \)\(97\!\cdots\!84\)\( p^{26} T^{10} - \)\(40\!\cdots\!84\)\( p^{39} T^{11} + \)\(90\!\cdots\!83\)\( p^{52} T^{12} + \)\(59\!\cdots\!56\)\( p^{65} T^{13} - \)\(58\!\cdots\!84\)\( p^{78} T^{14} + 81351201078 p^{91} T^{15} + p^{104} T^{16} \)
53 \( 1 - 87497947440 T - \)\(25\!\cdots\!70\)\( T^{2} - \)\(10\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!17\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} + \)\(61\!\cdots\!50\)\( T^{6} - \)\(92\!\cdots\!40\)\( T^{7} - \)\(12\!\cdots\!52\)\( T^{8} - \)\(92\!\cdots\!40\)\( p^{13} T^{9} + \)\(61\!\cdots\!50\)\( p^{26} T^{10} + \)\(23\!\cdots\!80\)\( p^{39} T^{11} + \)\(12\!\cdots\!17\)\( p^{52} T^{12} - \)\(10\!\cdots\!60\)\( p^{65} T^{13} - \)\(25\!\cdots\!70\)\( p^{78} T^{14} - 87497947440 p^{91} T^{15} + p^{104} T^{16} \)
59 \( 1 + 194140265102 T - \)\(32\!\cdots\!76\)\( T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(64\!\cdots\!63\)\( T^{4} + \)\(77\!\cdots\!04\)\( p T^{5} - \)\(95\!\cdots\!92\)\( T^{6} - \)\(19\!\cdots\!26\)\( T^{7} + \)\(11\!\cdots\!04\)\( T^{8} - \)\(19\!\cdots\!26\)\( p^{13} T^{9} - \)\(95\!\cdots\!92\)\( p^{26} T^{10} + \)\(77\!\cdots\!04\)\( p^{40} T^{11} + \)\(64\!\cdots\!63\)\( p^{52} T^{12} - \)\(39\!\cdots\!44\)\( p^{65} T^{13} - \)\(32\!\cdots\!76\)\( p^{78} T^{14} + 194140265102 p^{91} T^{15} + p^{104} T^{16} \)
61 \( 1 - 175816313120 T - \)\(34\!\cdots\!66\)\( T^{2} + \)\(45\!\cdots\!24\)\( T^{3} + \)\(61\!\cdots\!21\)\( T^{4} - \)\(52\!\cdots\!84\)\( T^{5} - \)\(53\!\cdots\!74\)\( T^{6} + \)\(38\!\cdots\!44\)\( T^{7} + \)\(34\!\cdots\!84\)\( T^{8} + \)\(38\!\cdots\!44\)\( p^{13} T^{9} - \)\(53\!\cdots\!74\)\( p^{26} T^{10} - \)\(52\!\cdots\!84\)\( p^{39} T^{11} + \)\(61\!\cdots\!21\)\( p^{52} T^{12} + \)\(45\!\cdots\!24\)\( p^{65} T^{13} - \)\(34\!\cdots\!66\)\( p^{78} T^{14} - 175816313120 p^{91} T^{15} + p^{104} T^{16} \)
67 \( 1 + 243815218758 T - \)\(13\!\cdots\!20\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(93\!\cdots\!15\)\( T^{4} - \)\(11\!\cdots\!12\)\( T^{5} - \)\(39\!\cdots\!60\)\( T^{6} + \)\(60\!\cdots\!54\)\( T^{7} + \)\(15\!\cdots\!76\)\( T^{8} + \)\(60\!\cdots\!54\)\( p^{13} T^{9} - \)\(39\!\cdots\!60\)\( p^{26} T^{10} - \)\(11\!\cdots\!12\)\( p^{39} T^{11} + \)\(93\!\cdots\!15\)\( p^{52} T^{12} - \)\(11\!\cdots\!40\)\( p^{65} T^{13} - \)\(13\!\cdots\!20\)\( p^{78} T^{14} + 243815218758 p^{91} T^{15} + p^{104} T^{16} \)
71 \( ( 1 + 1637697339536 T + \)\(40\!\cdots\!88\)\( T^{2} + \)\(37\!\cdots\!32\)\( T^{3} + \)\(59\!\cdots\!54\)\( T^{4} + \)\(37\!\cdots\!32\)\( p^{13} T^{5} + \)\(40\!\cdots\!88\)\( p^{26} T^{6} + 1637697339536 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
73 \( 1 + 3492491920596 T + \)\(28\!\cdots\!90\)\( T^{2} - \)\(48\!\cdots\!20\)\( T^{3} + \)\(38\!\cdots\!65\)\( T^{4} + \)\(58\!\cdots\!16\)\( T^{5} - \)\(55\!\cdots\!70\)\( T^{6} + \)\(74\!\cdots\!12\)\( T^{7} + \)\(37\!\cdots\!36\)\( T^{8} + \)\(74\!\cdots\!12\)\( p^{13} T^{9} - \)\(55\!\cdots\!70\)\( p^{26} T^{10} + \)\(58\!\cdots\!16\)\( p^{39} T^{11} + \)\(38\!\cdots\!65\)\( p^{52} T^{12} - \)\(48\!\cdots\!20\)\( p^{65} T^{13} + \)\(28\!\cdots\!90\)\( p^{78} T^{14} + 3492491920596 p^{91} T^{15} + p^{104} T^{16} \)
79 \( 1 + 1016380081246 T - \)\(18\!\cdots\!08\)\( T^{2} - \)\(55\!\cdots\!60\)\( T^{3} - \)\(37\!\cdots\!01\)\( T^{4} - \)\(52\!\cdots\!44\)\( T^{5} + \)\(39\!\cdots\!04\)\( T^{6} + \)\(22\!\cdots\!14\)\( T^{7} + \)\(11\!\cdots\!60\)\( T^{8} + \)\(22\!\cdots\!14\)\( p^{13} T^{9} + \)\(39\!\cdots\!04\)\( p^{26} T^{10} - \)\(52\!\cdots\!44\)\( p^{39} T^{11} - \)\(37\!\cdots\!01\)\( p^{52} T^{12} - \)\(55\!\cdots\!60\)\( p^{65} T^{13} - \)\(18\!\cdots\!08\)\( p^{78} T^{14} + 1016380081246 p^{91} T^{15} + p^{104} T^{16} \)
83 \( ( 1 - 3513747871648 T + \)\(13\!\cdots\!36\)\( T^{2} - \)\(57\!\cdots\!88\)\( p T^{3} + \)\(22\!\cdots\!98\)\( T^{4} - \)\(57\!\cdots\!88\)\( p^{14} T^{5} + \)\(13\!\cdots\!36\)\( p^{26} T^{6} - 3513747871648 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
89 \( 1 + 8034124428036 T + \)\(79\!\cdots\!34\)\( T^{2} + \)\(17\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!37\)\( T^{4} - \)\(59\!\cdots\!00\)\( T^{5} + \)\(84\!\cdots\!38\)\( T^{6} + \)\(62\!\cdots\!20\)\( p T^{7} - \)\(31\!\cdots\!12\)\( T^{8} + \)\(62\!\cdots\!20\)\( p^{14} T^{9} + \)\(84\!\cdots\!38\)\( p^{26} T^{10} - \)\(59\!\cdots\!00\)\( p^{39} T^{11} + \)\(13\!\cdots\!37\)\( p^{52} T^{12} + \)\(17\!\cdots\!36\)\( p^{65} T^{13} + \)\(79\!\cdots\!34\)\( p^{78} T^{14} + 8034124428036 p^{91} T^{15} + p^{104} T^{16} \)
97 \( ( 1 - 27175565862816 T + \)\(47\!\cdots\!36\)\( T^{2} - \)\(58\!\cdots\!56\)\( T^{3} + \)\(55\!\cdots\!86\)\( T^{4} - \)\(58\!\cdots\!56\)\( p^{13} T^{5} + \)\(47\!\cdots\!36\)\( p^{26} T^{6} - 27175565862816 p^{39} T^{7} + p^{52} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−6.63587916801606017246658549903, −6.50638570272043917978182658438, −6.05699737678555855878012955656, −6.04154215806733234124604822645, −5.89733127094827693787108809787, −4.96156213248973379642376791800, −4.91606615038611288384673916043, −4.70834082828787489641643198600, −4.39681338548090048843719704097, −4.33351481157201291120707662118, −4.10061134584600035771671004563, −3.89457206319466965417972139925, −3.42693116673649309312866110859, −3.14885543660382024191752635163, −3.03472302068902300112381514241, −2.51517502295507405607491982020, −2.38626363067284149308425982890, −1.80194600749434751019646227686, −1.48837903889266847969132186989, −1.41975547149108629559314415842, −0.994437450618950455823993121946, −0.959219610675506980943954305121, −0.48378893604349312030422742868, −0.44854618777649585499647510590, −0.27951950575615446141902666651, 0.27951950575615446141902666651, 0.44854618777649585499647510590, 0.48378893604349312030422742868, 0.959219610675506980943954305121, 0.994437450618950455823993121946, 1.41975547149108629559314415842, 1.48837903889266847969132186989, 1.80194600749434751019646227686, 2.38626363067284149308425982890, 2.51517502295507405607491982020, 3.03472302068902300112381514241, 3.14885543660382024191752635163, 3.42693116673649309312866110859, 3.89457206319466965417972139925, 4.10061134584600035771671004563, 4.33351481157201291120707662118, 4.39681338548090048843719704097, 4.70834082828787489641643198600, 4.91606615038611288384673916043, 4.96156213248973379642376791800, 5.89733127094827693787108809787, 6.04154215806733234124604822645, 6.05699737678555855878012955656, 6.50638570272043917978182658438, 6.63587916801606017246658549903

Graph of the $Z$-function along the critical line

Plot not available for L-functions of degree greater than 10.