# 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 of factors

## 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.