Properties

Label 16-14e8-1.1-c11e8-0-1
Degree $16$
Conductor $1475789056$
Sign $1$
Analytic cond. $1.79252\times 10^{8}$
Root an. cond. $3.27975$
Motivic weight $11$
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  + 128·2-s − 266·3-s + 6.14e3·4-s + 3.80e3·5-s − 3.40e4·6-s + 1.10e5·7-s + 1.37e5·9-s + 4.87e5·10-s + 9.20e5·11-s − 1.63e6·12-s + 9.97e5·13-s + 1.41e7·14-s − 1.01e6·15-s − 1.57e7·16-s + 1.33e6·17-s + 1.76e7·18-s − 2.15e7·19-s + 2.33e7·20-s − 2.93e7·21-s + 1.17e8·22-s + 7.25e7·23-s + 6.63e7·25-s + 1.27e8·26-s − 9.55e6·27-s + 6.77e8·28-s + 2.13e8·29-s − 1.29e8·30-s + ⋯
L(s)  = 1  + 2.82·2-s − 0.631·3-s + 3·4-s + 0.544·5-s − 1.78·6-s + 2.48·7-s + 0.777·9-s + 1.54·10-s + 1.72·11-s − 1.89·12-s + 0.745·13-s + 7.01·14-s − 0.344·15-s − 3.75·16-s + 0.227·17-s + 2.19·18-s − 1.99·19-s + 1.63·20-s − 1.56·21-s + 4.87·22-s + 2.34·23-s + 1.35·25-s + 2.10·26-s − 0.128·27-s + 7.44·28-s + 1.93·29-s − 0.974·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(16\)
Conductor: \(2^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(1.79252\times 10^{8}\)
Root analytic conductor: \(3.27975\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{14} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 2^{8} \cdot 7^{8} ,\ ( \ : [11/2]^{8} ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(82.14823654\)
\(L(\frac12)\) \(\approx\) \(82.14823654\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( ( 1 - p^{5} T + p^{10} T^{2} )^{4} \)
7 \( 1 - 110328 T + 1171736564 p T^{2} - 21942584040 p^{5} T^{3} + 424446026250 p^{9} T^{4} - 21942584040 p^{16} T^{5} + 1171736564 p^{23} T^{6} - 110328 p^{33} T^{7} + p^{44} T^{8} \)
good3 \( 1 + 266 T - 66992 T^{2} - 14969080 p T^{3} - 187789181 p^{2} T^{4} + 128977828 p^{10} T^{5} + 9648911678740 p^{6} T^{6} + 52785697751626 p^{9} T^{7} - 16793548592076368 p^{10} T^{8} + 52785697751626 p^{20} T^{9} + 9648911678740 p^{28} T^{10} + 128977828 p^{43} T^{11} - 187789181 p^{46} T^{12} - 14969080 p^{56} T^{13} - 66992 p^{66} T^{14} + 266 p^{77} T^{15} + p^{88} T^{16} \)
5 \( 1 - 3808 T - 51828446 T^{2} - 110199210576 p T^{3} + 144296540995493 p^{2} T^{4} + 220341431621070424 p^{3} T^{5} + \)\(44\!\cdots\!18\)\( p^{4} T^{6} - \)\(57\!\cdots\!36\)\( p^{5} T^{7} - \)\(77\!\cdots\!36\)\( p^{6} T^{8} - \)\(57\!\cdots\!36\)\( p^{16} T^{9} + \)\(44\!\cdots\!18\)\( p^{26} T^{10} + 220341431621070424 p^{36} T^{11} + 144296540995493 p^{46} T^{12} - 110199210576 p^{56} T^{13} - 51828446 p^{66} T^{14} - 3808 p^{77} T^{15} + p^{88} T^{16} \)
11 \( 1 - 83650 p T - 402713996672 T^{2} + 2510100354409368 p^{2} T^{3} + \)\(35\!\cdots\!67\)\( T^{4} - \)\(99\!\cdots\!72\)\( p^{2} T^{5} - \)\(13\!\cdots\!04\)\( T^{6} + \)\(92\!\cdots\!58\)\( p T^{7} + \)\(47\!\cdots\!28\)\( T^{8} + \)\(92\!\cdots\!58\)\( p^{12} T^{9} - \)\(13\!\cdots\!04\)\( p^{22} T^{10} - \)\(99\!\cdots\!72\)\( p^{35} T^{11} + \)\(35\!\cdots\!67\)\( p^{44} T^{12} + 2510100354409368 p^{57} T^{13} - 402713996672 p^{66} T^{14} - 83650 p^{78} T^{15} + p^{88} T^{16} \)
13 \( ( 1 - 498736 T + 3114010681004 T^{2} - 929649607137739152 T^{3} + \)\(71\!\cdots\!98\)\( T^{4} - 929649607137739152 p^{11} T^{5} + 3114010681004 p^{22} T^{6} - 498736 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
17 \( 1 - 1333724 T - 84999404037266 T^{2} + \)\(11\!\cdots\!08\)\( T^{3} + \)\(34\!\cdots\!33\)\( T^{4} - \)\(34\!\cdots\!28\)\( T^{5} - \)\(12\!\cdots\!34\)\( T^{6} + \)\(44\!\cdots\!64\)\( T^{7} + \)\(46\!\cdots\!16\)\( T^{8} + \)\(44\!\cdots\!64\)\( p^{11} T^{9} - \)\(12\!\cdots\!34\)\( p^{22} T^{10} - \)\(34\!\cdots\!28\)\( p^{33} T^{11} + \)\(34\!\cdots\!33\)\( p^{44} T^{12} + \)\(11\!\cdots\!08\)\( p^{55} T^{13} - 84999404037266 p^{66} T^{14} - 1333724 p^{77} T^{15} + p^{88} T^{16} \)
19 \( 1 + 21551726 T - 35225959518816 T^{2} - \)\(39\!\cdots\!24\)\( T^{3} + \)\(20\!\cdots\!27\)\( T^{4} + \)\(67\!\cdots\!80\)\( T^{5} + \)\(43\!\cdots\!68\)\( T^{6} - \)\(18\!\cdots\!90\)\( T^{7} - \)\(39\!\cdots\!32\)\( T^{8} - \)\(18\!\cdots\!90\)\( p^{11} T^{9} + \)\(43\!\cdots\!68\)\( p^{22} T^{10} + \)\(67\!\cdots\!80\)\( p^{33} T^{11} + \)\(20\!\cdots\!27\)\( p^{44} T^{12} - \)\(39\!\cdots\!24\)\( p^{55} T^{13} - 35225959518816 p^{66} T^{14} + 21551726 p^{77} T^{15} + p^{88} T^{16} \)
23 \( 1 - 72510158 T + 850551352355164 T^{2} + \)\(94\!\cdots\!12\)\( T^{3} + \)\(15\!\cdots\!19\)\( T^{4} - \)\(19\!\cdots\!84\)\( T^{5} - \)\(25\!\cdots\!60\)\( T^{6} + \)\(50\!\cdots\!38\)\( T^{7} + \)\(19\!\cdots\!00\)\( T^{8} + \)\(50\!\cdots\!38\)\( p^{11} T^{9} - \)\(25\!\cdots\!60\)\( p^{22} T^{10} - \)\(19\!\cdots\!84\)\( p^{33} T^{11} + \)\(15\!\cdots\!19\)\( p^{44} T^{12} + \)\(94\!\cdots\!12\)\( p^{55} T^{13} + 850551352355164 p^{66} T^{14} - 72510158 p^{77} T^{15} + p^{88} T^{16} \)
29 \( ( 1 - 106787552 T + 14829553290576972 T^{2} + \)\(15\!\cdots\!28\)\( T^{3} - \)\(14\!\cdots\!38\)\( T^{4} + \)\(15\!\cdots\!28\)\( p^{11} T^{5} + 14829553290576972 p^{22} T^{6} - 106787552 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
31 \( 1 + 194359774 T - 55172562054970420 T^{2} - \)\(60\!\cdots\!44\)\( T^{3} + \)\(29\!\cdots\!15\)\( T^{4} + \)\(11\!\cdots\!48\)\( T^{5} - \)\(10\!\cdots\!68\)\( T^{6} - \)\(15\!\cdots\!70\)\( T^{7} + \)\(28\!\cdots\!04\)\( T^{8} - \)\(15\!\cdots\!70\)\( p^{11} T^{9} - \)\(10\!\cdots\!68\)\( p^{22} T^{10} + \)\(11\!\cdots\!48\)\( p^{33} T^{11} + \)\(29\!\cdots\!15\)\( p^{44} T^{12} - \)\(60\!\cdots\!44\)\( p^{55} T^{13} - 55172562054970420 p^{66} T^{14} + 194359774 p^{77} T^{15} + p^{88} T^{16} \)
37 \( 1 - 171517048 T + 144669798644744010 T^{2} + \)\(92\!\cdots\!08\)\( T^{3} - \)\(46\!\cdots\!71\)\( T^{4} + \)\(16\!\cdots\!72\)\( T^{5} - \)\(85\!\cdots\!06\)\( T^{6} - \)\(37\!\cdots\!64\)\( T^{7} + \)\(15\!\cdots\!60\)\( T^{8} - \)\(37\!\cdots\!64\)\( p^{11} T^{9} - \)\(85\!\cdots\!06\)\( p^{22} T^{10} + \)\(16\!\cdots\!72\)\( p^{33} T^{11} - \)\(46\!\cdots\!71\)\( p^{44} T^{12} + \)\(92\!\cdots\!08\)\( p^{55} T^{13} + 144669798644744010 p^{66} T^{14} - 171517048 p^{77} T^{15} + p^{88} T^{16} \)
41 \( ( 1 + 1656047568 T + 2530400059494857852 T^{2} + \)\(23\!\cdots\!52\)\( T^{3} + \)\(20\!\cdots\!54\)\( T^{4} + \)\(23\!\cdots\!52\)\( p^{11} T^{5} + 2530400059494857852 p^{22} T^{6} + 1656047568 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
43 \( ( 1 - 425139824 T + 3462234885788574892 T^{2} - \)\(10\!\cdots\!80\)\( T^{3} + \)\(47\!\cdots\!50\)\( T^{4} - \)\(10\!\cdots\!80\)\( p^{11} T^{5} + 3462234885788574892 p^{22} T^{6} - 425139824 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
47 \( 1 - 2223880974 T - 4753526137652652356 T^{2} + \)\(81\!\cdots\!48\)\( T^{3} + \)\(25\!\cdots\!63\)\( T^{4} - \)\(22\!\cdots\!68\)\( T^{5} - \)\(87\!\cdots\!44\)\( T^{6} + \)\(38\!\cdots\!42\)\( p T^{7} + \)\(11\!\cdots\!64\)\( p^{2} T^{8} + \)\(38\!\cdots\!42\)\( p^{12} T^{9} - \)\(87\!\cdots\!44\)\( p^{22} T^{10} - \)\(22\!\cdots\!68\)\( p^{33} T^{11} + \)\(25\!\cdots\!63\)\( p^{44} T^{12} + \)\(81\!\cdots\!48\)\( p^{55} T^{13} - 4753526137652652356 p^{66} T^{14} - 2223880974 p^{77} T^{15} + p^{88} T^{16} \)
53 \( 1 + 7185483360 T + 7627134222160004290 T^{2} - \)\(65\!\cdots\!40\)\( T^{3} - \)\(69\!\cdots\!43\)\( T^{4} + \)\(10\!\cdots\!80\)\( T^{5} + \)\(64\!\cdots\!50\)\( p T^{6} - \)\(18\!\cdots\!60\)\( T^{7} - \)\(27\!\cdots\!32\)\( T^{8} - \)\(18\!\cdots\!60\)\( p^{11} T^{9} + \)\(64\!\cdots\!50\)\( p^{23} T^{10} + \)\(10\!\cdots\!80\)\( p^{33} T^{11} - \)\(69\!\cdots\!43\)\( p^{44} T^{12} - \)\(65\!\cdots\!40\)\( p^{55} T^{13} + 7627134222160004290 p^{66} T^{14} + 7185483360 p^{77} T^{15} + p^{88} T^{16} \)
59 \( 1 - 6997401502 T - 41409964972928294216 T^{2} + \)\(42\!\cdots\!04\)\( T^{3} + \)\(69\!\cdots\!83\)\( T^{4} - \)\(11\!\cdots\!96\)\( T^{5} - \)\(75\!\cdots\!32\)\( T^{6} + \)\(12\!\cdots\!66\)\( T^{7} + \)\(17\!\cdots\!44\)\( T^{8} + \)\(12\!\cdots\!66\)\( p^{11} T^{9} - \)\(75\!\cdots\!32\)\( p^{22} T^{10} - \)\(11\!\cdots\!96\)\( p^{33} T^{11} + \)\(69\!\cdots\!83\)\( p^{44} T^{12} + \)\(42\!\cdots\!04\)\( p^{55} T^{13} - 41409964972928294216 p^{66} T^{14} - 6997401502 p^{77} T^{15} + p^{88} T^{16} \)
61 \( 1 + 6476463280 T - 87626038568336031646 T^{2} - \)\(23\!\cdots\!56\)\( T^{3} + \)\(60\!\cdots\!01\)\( T^{4} - \)\(95\!\cdots\!24\)\( T^{5} - \)\(31\!\cdots\!34\)\( T^{6} - \)\(36\!\cdots\!56\)\( T^{7} + \)\(11\!\cdots\!84\)\( T^{8} - \)\(36\!\cdots\!56\)\( p^{11} T^{9} - \)\(31\!\cdots\!34\)\( p^{22} T^{10} - \)\(95\!\cdots\!24\)\( p^{33} T^{11} + \)\(60\!\cdots\!01\)\( p^{44} T^{12} - \)\(23\!\cdots\!56\)\( p^{55} T^{13} - 87626038568336031646 p^{66} T^{14} + 6476463280 p^{77} T^{15} + p^{88} T^{16} \)
67 \( 1 + 18660972186 T - \)\(24\!\cdots\!20\)\( T^{2} - \)\(30\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!55\)\( T^{4} + \)\(74\!\cdots\!76\)\( T^{5} - \)\(16\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!18\)\( T^{7} + \)\(29\!\cdots\!76\)\( T^{8} - \)\(10\!\cdots\!18\)\( p^{11} T^{9} - \)\(16\!\cdots\!20\)\( p^{22} T^{10} + \)\(74\!\cdots\!76\)\( p^{33} T^{11} + \)\(10\!\cdots\!55\)\( p^{44} T^{12} - \)\(30\!\cdots\!80\)\( p^{55} T^{13} - \)\(24\!\cdots\!20\)\( p^{66} T^{14} + 18660972186 p^{77} T^{15} + p^{88} T^{16} \)
71 \( ( 1 - 11612224624 T + \)\(85\!\cdots\!08\)\( T^{2} - \)\(81\!\cdots\!48\)\( T^{3} + \)\(28\!\cdots\!94\)\( T^{4} - \)\(81\!\cdots\!48\)\( p^{11} T^{5} + \)\(85\!\cdots\!08\)\( p^{22} T^{6} - 11612224624 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
73 \( 1 - 3731641452 T - \)\(73\!\cdots\!50\)\( T^{2} - \)\(45\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!45\)\( T^{4} + \)\(29\!\cdots\!48\)\( T^{5} - \)\(52\!\cdots\!30\)\( T^{6} - \)\(57\!\cdots\!16\)\( T^{7} + \)\(75\!\cdots\!36\)\( T^{8} - \)\(57\!\cdots\!16\)\( p^{11} T^{9} - \)\(52\!\cdots\!30\)\( p^{22} T^{10} + \)\(29\!\cdots\!48\)\( p^{33} T^{11} + \)\(30\!\cdots\!45\)\( p^{44} T^{12} - \)\(45\!\cdots\!60\)\( p^{55} T^{13} - \)\(73\!\cdots\!50\)\( p^{66} T^{14} - 3731641452 p^{77} T^{15} + p^{88} T^{16} \)
79 \( 1 - 12221157926 T - \)\(11\!\cdots\!28\)\( T^{2} + \)\(97\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!99\)\( T^{4} + \)\(74\!\cdots\!64\)\( T^{5} - \)\(36\!\cdots\!76\)\( T^{6} - \)\(66\!\cdots\!34\)\( T^{7} + \)\(71\!\cdots\!60\)\( T^{8} - \)\(66\!\cdots\!34\)\( p^{11} T^{9} - \)\(36\!\cdots\!76\)\( p^{22} T^{10} + \)\(74\!\cdots\!64\)\( p^{33} T^{11} + \)\(22\!\cdots\!99\)\( p^{44} T^{12} + \)\(97\!\cdots\!80\)\( p^{55} T^{13} - \)\(11\!\cdots\!28\)\( p^{66} T^{14} - 12221157926 p^{77} T^{15} + p^{88} T^{16} \)
83 \( ( 1 - 158369761984 T + \)\(13\!\cdots\!44\)\( T^{2} - \)\(76\!\cdots\!28\)\( T^{3} + \)\(31\!\cdots\!18\)\( T^{4} - \)\(76\!\cdots\!28\)\( p^{11} T^{5} + \)\(13\!\cdots\!44\)\( p^{22} T^{6} - 158369761984 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
89 \( 1 + 71204406084 T - \)\(74\!\cdots\!06\)\( T^{2} - \)\(29\!\cdots\!36\)\( T^{3} + \)\(60\!\cdots\!77\)\( T^{4} + \)\(14\!\cdots\!80\)\( T^{5} - \)\(24\!\cdots\!62\)\( T^{6} - \)\(84\!\cdots\!60\)\( T^{7} + \)\(89\!\cdots\!48\)\( T^{8} - \)\(84\!\cdots\!60\)\( p^{11} T^{9} - \)\(24\!\cdots\!62\)\( p^{22} T^{10} + \)\(14\!\cdots\!80\)\( p^{33} T^{11} + \)\(60\!\cdots\!77\)\( p^{44} T^{12} - \)\(29\!\cdots\!36\)\( p^{55} T^{13} - \)\(74\!\cdots\!06\)\( p^{66} T^{14} + 71204406084 p^{77} T^{15} + p^{88} T^{16} \)
97 \( ( 1 - 344125898592 T + \)\(63\!\cdots\!04\)\( T^{2} - \)\(78\!\cdots\!08\)\( T^{3} + \)\(74\!\cdots\!06\)\( T^{4} - \)\(78\!\cdots\!08\)\( p^{11} T^{5} + \)\(63\!\cdots\!04\)\( p^{22} T^{6} - 344125898592 p^{33} T^{7} + p^{44} T^{8} )^{2} \)
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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

−6.64943066780886379378273731135, −6.39728241996441805596361567565, −6.28704532519948676142195755053, −6.28041319361112453481193117277, −6.21585976389065476653017788690, −5.29979912456155123835274000647, −5.24801157541348561365364648985, −5.02836229943908090349208410838, −4.95953813548979803806025480965, −4.85666899907774975184667146611, −4.65757345000322346618800266234, −4.32109984959204031239102045699, −3.97351810514835812246180015436, −3.80709711053136351382600233268, −3.40842652158347037021910907317, −3.36066894178720064009573345957, −3.01337058184945248721127549766, −2.48000016799933002580070895472, −2.08398557054704233137020707721, −1.87480216840658501293542927807, −1.57620362677382716562257570701, −1.33312102960491829029000514300, −0.893406218011318205218746597811, −0.59310987018229284807386746760, −0.53338866246804771731124828469, 0.53338866246804771731124828469, 0.59310987018229284807386746760, 0.893406218011318205218746597811, 1.33312102960491829029000514300, 1.57620362677382716562257570701, 1.87480216840658501293542927807, 2.08398557054704233137020707721, 2.48000016799933002580070895472, 3.01337058184945248721127549766, 3.36066894178720064009573345957, 3.40842652158347037021910907317, 3.80709711053136351382600233268, 3.97351810514835812246180015436, 4.32109984959204031239102045699, 4.65757345000322346618800266234, 4.85666899907774975184667146611, 4.95953813548979803806025480965, 5.02836229943908090349208410838, 5.24801157541348561365364648985, 5.29979912456155123835274000647, 6.21585976389065476653017788690, 6.28041319361112453481193117277, 6.28704532519948676142195755053, 6.39728241996441805596361567565, 6.64943066780886379378273731135

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

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