Properties

Label 18-1-1.1-c121e9-0-0
Degree $18$
Conductor $1$
Sign $-1$
Analytic cond. $5.06346\times 10^{17}$
Root an. cond. $9.62898$
Motivic weight $121$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $9$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.34e18·2-s − 4.52e28·3-s − 4.98e36·4-s − 1.89e42·5-s + 1.06e47·6-s + 2.19e51·7-s + 1.58e55·8-s − 1.92e58·9-s + 4.43e60·10-s − 9.46e62·11-s + 2.25e65·12-s + 2.05e67·13-s − 5.14e69·14-s + 8.57e70·15-s + 8.25e72·16-s + 2.91e74·17-s + 4.51e76·18-s + 3.02e76·19-s + 9.42e78·20-s − 9.94e79·21-s + 2.21e81·22-s − 6.65e82·23-s − 7.19e83·24-s − 1.81e85·25-s − 4.82e85·26-s + 7.85e86·27-s − 1.09e88·28-s + ⋯
L(s)  = 1  − 1.43·2-s − 0.616·3-s − 1.87·4-s − 0.975·5-s + 0.887·6-s + 1.63·7-s + 3.66·8-s − 3.57·9-s + 1.40·10-s − 0.937·11-s + 1.15·12-s + 0.832·13-s − 2.34·14-s + 0.602·15-s + 1.16·16-s + 1.05·17-s + 5.13·18-s + 0.130·19-s + 1.82·20-s − 1.00·21-s + 1.34·22-s − 2.74·23-s − 2.26·24-s − 4.82·25-s − 1.19·26-s + 1.98·27-s − 3.06·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(122-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+60.5)^{9} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(18\)
Conductor: \(1\)
Sign: $-1$
Analytic conductor: \(5.06346\times 10^{17}\)
Root analytic conductor: \(9.62898\)
Motivic weight: \(121\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(9\)
Selberg data: \((18,\ 1,\ (\ :[121/2]^{9}),\ -1)\)

Particular Values

\(L(61)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{123}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 + 146562847640907483 p^{4} T + \)\(63\!\cdots\!73\)\( p^{14} T^{2} + \)\(30\!\cdots\!35\)\( p^{26} T^{3} + \)\(49\!\cdots\!01\)\( p^{40} T^{4} + \)\(20\!\cdots\!37\)\( p^{62} T^{5} + \)\(62\!\cdots\!57\)\( p^{88} T^{6} + \)\(40\!\cdots\!55\)\( p^{116} T^{7} + \)\(64\!\cdots\!19\)\( p^{146} T^{8} + \)\(63\!\cdots\!63\)\( p^{180} T^{9} + \)\(64\!\cdots\!19\)\( p^{267} T^{10} + \)\(40\!\cdots\!55\)\( p^{358} T^{11} + \)\(62\!\cdots\!57\)\( p^{451} T^{12} + \)\(20\!\cdots\!37\)\( p^{546} T^{13} + \)\(49\!\cdots\!01\)\( p^{645} T^{14} + \)\(30\!\cdots\!35\)\( p^{752} T^{15} + \)\(63\!\cdots\!73\)\( p^{861} T^{16} + 146562847640907483 p^{972} T^{17} + p^{1089} T^{18} \)
3 \( 1 + \)\(18\!\cdots\!28\)\( p^{5} T + \)\(12\!\cdots\!29\)\( p^{11} T^{2} + \)\(27\!\cdots\!60\)\( p^{18} T^{3} + \)\(40\!\cdots\!28\)\( p^{31} T^{4} + \)\(14\!\cdots\!36\)\( p^{46} T^{5} + \)\(64\!\cdots\!88\)\( p^{64} T^{6} + \)\(99\!\cdots\!20\)\( p^{86} T^{7} + \)\(16\!\cdots\!18\)\( p^{111} T^{8} + \)\(93\!\cdots\!96\)\( p^{138} T^{9} + \)\(16\!\cdots\!18\)\( p^{232} T^{10} + \)\(99\!\cdots\!20\)\( p^{328} T^{11} + \)\(64\!\cdots\!88\)\( p^{427} T^{12} + \)\(14\!\cdots\!36\)\( p^{530} T^{13} + \)\(40\!\cdots\!28\)\( p^{636} T^{14} + \)\(27\!\cdots\!60\)\( p^{744} T^{15} + \)\(12\!\cdots\!29\)\( p^{858} T^{16} + \)\(18\!\cdots\!28\)\( p^{973} T^{17} + p^{1089} T^{18} \)
5 \( 1 + \)\(75\!\cdots\!62\)\( p^{2} T + \)\(55\!\cdots\!81\)\( p^{8} T^{2} + \)\(15\!\cdots\!88\)\( p^{15} T^{3} + \)\(15\!\cdots\!12\)\( p^{26} T^{4} + \)\(18\!\cdots\!84\)\( p^{35} T^{5} + \)\(53\!\cdots\!28\)\( p^{45} T^{6} + \)\(10\!\cdots\!44\)\( p^{58} T^{7} + \)\(35\!\cdots\!02\)\( p^{72} T^{8} + \)\(39\!\cdots\!96\)\( p^{91} T^{9} + \)\(35\!\cdots\!02\)\( p^{193} T^{10} + \)\(10\!\cdots\!44\)\( p^{300} T^{11} + \)\(53\!\cdots\!28\)\( p^{408} T^{12} + \)\(18\!\cdots\!84\)\( p^{519} T^{13} + \)\(15\!\cdots\!12\)\( p^{631} T^{14} + \)\(15\!\cdots\!88\)\( p^{741} T^{15} + \)\(55\!\cdots\!81\)\( p^{855} T^{16} + \)\(75\!\cdots\!62\)\( p^{970} T^{17} + p^{1089} T^{18} \)
7 \( 1 - \)\(44\!\cdots\!08\)\( p^{2} T + \)\(50\!\cdots\!01\)\( p^{5} T^{2} - \)\(48\!\cdots\!00\)\( p^{10} T^{3} + \)\(13\!\cdots\!28\)\( p^{17} T^{4} - \)\(28\!\cdots\!76\)\( p^{25} T^{5} + \)\(12\!\cdots\!28\)\( p^{34} T^{6} - \)\(65\!\cdots\!00\)\( p^{45} T^{7} + \)\(11\!\cdots\!94\)\( p^{58} T^{8} - \)\(16\!\cdots\!52\)\( p^{72} T^{9} + \)\(11\!\cdots\!94\)\( p^{179} T^{10} - \)\(65\!\cdots\!00\)\( p^{287} T^{11} + \)\(12\!\cdots\!28\)\( p^{397} T^{12} - \)\(28\!\cdots\!76\)\( p^{509} T^{13} + \)\(13\!\cdots\!28\)\( p^{622} T^{14} - \)\(48\!\cdots\!00\)\( p^{736} T^{15} + \)\(50\!\cdots\!01\)\( p^{852} T^{16} - \)\(44\!\cdots\!08\)\( p^{970} T^{17} + p^{1089} T^{18} \)
11 \( 1 + \)\(94\!\cdots\!92\)\( T + \)\(39\!\cdots\!93\)\( p^{3} T^{2} + \)\(28\!\cdots\!64\)\( p^{7} T^{3} + \)\(49\!\cdots\!60\)\( p^{12} T^{4} + \)\(28\!\cdots\!92\)\( p^{17} T^{5} + \)\(36\!\cdots\!04\)\( p^{22} T^{6} + \)\(19\!\cdots\!16\)\( p^{27} T^{7} + \)\(17\!\cdots\!38\)\( p^{33} T^{8} + \)\(68\!\cdots\!80\)\( p^{40} T^{9} + \)\(17\!\cdots\!38\)\( p^{154} T^{10} + \)\(19\!\cdots\!16\)\( p^{269} T^{11} + \)\(36\!\cdots\!04\)\( p^{385} T^{12} + \)\(28\!\cdots\!92\)\( p^{501} T^{13} + \)\(49\!\cdots\!60\)\( p^{617} T^{14} + \)\(28\!\cdots\!64\)\( p^{733} T^{15} + \)\(39\!\cdots\!93\)\( p^{850} T^{16} + \)\(94\!\cdots\!92\)\( p^{968} T^{17} + p^{1089} T^{18} \)
13 \( 1 - \)\(15\!\cdots\!22\)\( p T + \)\(13\!\cdots\!89\)\( p^{3} T^{2} - \)\(15\!\cdots\!80\)\( p^{6} T^{3} + \)\(33\!\cdots\!64\)\( p^{10} T^{4} - \)\(24\!\cdots\!48\)\( p^{15} T^{5} + \)\(19\!\cdots\!56\)\( p^{21} T^{6} - \)\(11\!\cdots\!20\)\( p^{27} T^{7} + \)\(64\!\cdots\!62\)\( p^{33} T^{8} - \)\(27\!\cdots\!76\)\( p^{40} T^{9} + \)\(64\!\cdots\!62\)\( p^{154} T^{10} - \)\(11\!\cdots\!20\)\( p^{269} T^{11} + \)\(19\!\cdots\!56\)\( p^{384} T^{12} - \)\(24\!\cdots\!48\)\( p^{499} T^{13} + \)\(33\!\cdots\!64\)\( p^{615} T^{14} - \)\(15\!\cdots\!80\)\( p^{732} T^{15} + \)\(13\!\cdots\!89\)\( p^{850} T^{16} - \)\(15\!\cdots\!22\)\( p^{969} T^{17} + p^{1089} T^{18} \)
17 \( 1 - \)\(17\!\cdots\!46\)\( p T + \)\(81\!\cdots\!89\)\( p^{3} T^{2} - \)\(88\!\cdots\!40\)\( p^{5} T^{3} + \)\(11\!\cdots\!96\)\( p^{8} T^{4} - \)\(42\!\cdots\!72\)\( p^{12} T^{5} + \)\(24\!\cdots\!72\)\( p^{16} T^{6} - \)\(75\!\cdots\!60\)\( p^{20} T^{7} + \)\(35\!\cdots\!66\)\( p^{24} T^{8} - \)\(56\!\cdots\!56\)\( p^{29} T^{9} + \)\(35\!\cdots\!66\)\( p^{145} T^{10} - \)\(75\!\cdots\!60\)\( p^{262} T^{11} + \)\(24\!\cdots\!72\)\( p^{379} T^{12} - \)\(42\!\cdots\!72\)\( p^{496} T^{13} + \)\(11\!\cdots\!96\)\( p^{613} T^{14} - \)\(88\!\cdots\!40\)\( p^{731} T^{15} + \)\(81\!\cdots\!89\)\( p^{850} T^{16} - \)\(17\!\cdots\!46\)\( p^{969} T^{17} + p^{1089} T^{18} \)
19 \( 1 - \)\(15\!\cdots\!20\)\( p T + \)\(63\!\cdots\!11\)\( p^{2} T^{2} - \)\(21\!\cdots\!60\)\( p^{4} T^{3} + \)\(31\!\cdots\!64\)\( p^{7} T^{4} - \)\(75\!\cdots\!40\)\( p^{10} T^{5} + \)\(33\!\cdots\!36\)\( p^{14} T^{6} - \)\(21\!\cdots\!80\)\( p^{19} T^{7} + \)\(38\!\cdots\!26\)\( p^{24} T^{8} - \)\(11\!\cdots\!00\)\( p^{30} T^{9} + \)\(38\!\cdots\!26\)\( p^{145} T^{10} - \)\(21\!\cdots\!80\)\( p^{261} T^{11} + \)\(33\!\cdots\!36\)\( p^{377} T^{12} - \)\(75\!\cdots\!40\)\( p^{494} T^{13} + \)\(31\!\cdots\!64\)\( p^{612} T^{14} - \)\(21\!\cdots\!60\)\( p^{730} T^{15} + \)\(63\!\cdots\!11\)\( p^{849} T^{16} - \)\(15\!\cdots\!20\)\( p^{969} T^{17} + p^{1089} T^{18} \)
23 \( 1 + \)\(66\!\cdots\!24\)\( T + \)\(20\!\cdots\!61\)\( p T^{2} + \)\(16\!\cdots\!60\)\( p^{3} T^{3} + \)\(12\!\cdots\!92\)\( p^{5} T^{4} + \)\(30\!\cdots\!64\)\( p^{8} T^{5} + \)\(70\!\cdots\!64\)\( p^{11} T^{6} + \)\(61\!\cdots\!20\)\( p^{15} T^{7} + \)\(52\!\cdots\!98\)\( p^{19} T^{8} + \)\(19\!\cdots\!44\)\( p^{24} T^{9} + \)\(52\!\cdots\!98\)\( p^{140} T^{10} + \)\(61\!\cdots\!20\)\( p^{257} T^{11} + \)\(70\!\cdots\!64\)\( p^{374} T^{12} + \)\(30\!\cdots\!64\)\( p^{492} T^{13} + \)\(12\!\cdots\!92\)\( p^{610} T^{14} + \)\(16\!\cdots\!60\)\( p^{729} T^{15} + \)\(20\!\cdots\!61\)\( p^{848} T^{16} + \)\(66\!\cdots\!24\)\( p^{968} T^{17} + p^{1089} T^{18} \)
29 \( 1 - \)\(31\!\cdots\!30\)\( p T + \)\(47\!\cdots\!21\)\( p^{2} T^{2} - \)\(83\!\cdots\!40\)\( p^{4} T^{3} + \)\(13\!\cdots\!56\)\( p^{6} T^{4} - \)\(33\!\cdots\!60\)\( p^{8} T^{5} + \)\(95\!\cdots\!44\)\( p^{11} T^{6} - \)\(90\!\cdots\!80\)\( p^{14} T^{7} + \)\(60\!\cdots\!46\)\( p^{18} T^{8} - \)\(19\!\cdots\!00\)\( p^{22} T^{9} + \)\(60\!\cdots\!46\)\( p^{139} T^{10} - \)\(90\!\cdots\!80\)\( p^{256} T^{11} + \)\(95\!\cdots\!44\)\( p^{374} T^{12} - \)\(33\!\cdots\!60\)\( p^{492} T^{13} + \)\(13\!\cdots\!56\)\( p^{611} T^{14} - \)\(83\!\cdots\!40\)\( p^{730} T^{15} + \)\(47\!\cdots\!21\)\( p^{849} T^{16} - \)\(31\!\cdots\!30\)\( p^{969} T^{17} + p^{1089} T^{18} \)
31 \( 1 + \)\(10\!\cdots\!12\)\( T + \)\(15\!\cdots\!43\)\( T^{2} + \)\(49\!\cdots\!44\)\( p T^{3} + \)\(11\!\cdots\!60\)\( p^{2} T^{4} + \)\(12\!\cdots\!92\)\( p^{4} T^{5} + \)\(58\!\cdots\!04\)\( p^{6} T^{6} + \)\(64\!\cdots\!16\)\( p^{8} T^{7} + \)\(72\!\cdots\!78\)\( p^{11} T^{8} + \)\(24\!\cdots\!80\)\( p^{14} T^{9} + \)\(72\!\cdots\!78\)\( p^{132} T^{10} + \)\(64\!\cdots\!16\)\( p^{250} T^{11} + \)\(58\!\cdots\!04\)\( p^{369} T^{12} + \)\(12\!\cdots\!92\)\( p^{488} T^{13} + \)\(11\!\cdots\!60\)\( p^{607} T^{14} + \)\(49\!\cdots\!44\)\( p^{727} T^{15} + \)\(15\!\cdots\!43\)\( p^{847} T^{16} + \)\(10\!\cdots\!12\)\( p^{968} T^{17} + p^{1089} T^{18} \)
37 \( 1 + \)\(23\!\cdots\!74\)\( p T + \)\(21\!\cdots\!53\)\( p^{2} T^{2} + \)\(33\!\cdots\!20\)\( p^{3} T^{3} + \)\(52\!\cdots\!88\)\( p^{5} T^{4} + \)\(17\!\cdots\!36\)\( p^{7} T^{5} + \)\(24\!\cdots\!76\)\( p^{9} T^{6} + \)\(19\!\cdots\!20\)\( p^{12} T^{7} + \)\(71\!\cdots\!22\)\( p^{15} T^{8} + \)\(52\!\cdots\!52\)\( p^{18} T^{9} + \)\(71\!\cdots\!22\)\( p^{136} T^{10} + \)\(19\!\cdots\!20\)\( p^{254} T^{11} + \)\(24\!\cdots\!76\)\( p^{372} T^{12} + \)\(17\!\cdots\!36\)\( p^{491} T^{13} + \)\(52\!\cdots\!88\)\( p^{610} T^{14} + \)\(33\!\cdots\!20\)\( p^{729} T^{15} + \)\(21\!\cdots\!53\)\( p^{849} T^{16} + \)\(23\!\cdots\!74\)\( p^{969} T^{17} + p^{1089} T^{18} \)
41 \( 1 + \)\(36\!\cdots\!22\)\( T + \)\(60\!\cdots\!73\)\( T^{2} + \)\(24\!\cdots\!24\)\( T^{3} + \)\(54\!\cdots\!60\)\( p T^{4} + \)\(48\!\cdots\!72\)\( p^{2} T^{5} + \)\(81\!\cdots\!64\)\( p^{3} T^{6} + \)\(15\!\cdots\!16\)\( p^{5} T^{7} + \)\(53\!\cdots\!98\)\( p^{7} T^{8} + \)\(92\!\cdots\!80\)\( p^{9} T^{9} + \)\(53\!\cdots\!98\)\( p^{128} T^{10} + \)\(15\!\cdots\!16\)\( p^{247} T^{11} + \)\(81\!\cdots\!64\)\( p^{366} T^{12} + \)\(48\!\cdots\!72\)\( p^{486} T^{13} + \)\(54\!\cdots\!60\)\( p^{606} T^{14} + \)\(24\!\cdots\!24\)\( p^{726} T^{15} + \)\(60\!\cdots\!73\)\( p^{847} T^{16} + \)\(36\!\cdots\!22\)\( p^{968} T^{17} + p^{1089} T^{18} \)
43 \( 1 + \)\(23\!\cdots\!44\)\( T + \)\(29\!\cdots\!43\)\( T^{2} + \)\(13\!\cdots\!00\)\( p T^{3} + \)\(22\!\cdots\!04\)\( p^{2} T^{4} + \)\(92\!\cdots\!32\)\( p^{3} T^{5} + \)\(11\!\cdots\!28\)\( p^{4} T^{6} + \)\(90\!\cdots\!00\)\( p^{6} T^{7} + \)\(20\!\cdots\!06\)\( p^{8} T^{8} + \)\(14\!\cdots\!36\)\( p^{10} T^{9} + \)\(20\!\cdots\!06\)\( p^{129} T^{10} + \)\(90\!\cdots\!00\)\( p^{248} T^{11} + \)\(11\!\cdots\!28\)\( p^{367} T^{12} + \)\(92\!\cdots\!32\)\( p^{487} T^{13} + \)\(22\!\cdots\!04\)\( p^{607} T^{14} + \)\(13\!\cdots\!00\)\( p^{727} T^{15} + \)\(29\!\cdots\!43\)\( p^{847} T^{16} + \)\(23\!\cdots\!44\)\( p^{968} T^{17} + p^{1089} T^{18} \)
47 \( 1 + \)\(13\!\cdots\!48\)\( T + \)\(10\!\cdots\!07\)\( T^{2} + \)\(17\!\cdots\!40\)\( p T^{3} + \)\(21\!\cdots\!84\)\( p^{2} T^{4} + \)\(94\!\cdots\!36\)\( p^{3} T^{5} + \)\(25\!\cdots\!52\)\( p^{4} T^{6} - \)\(38\!\cdots\!60\)\( p^{6} T^{7} + \)\(10\!\cdots\!66\)\( p^{8} T^{8} - \)\(31\!\cdots\!28\)\( p^{10} T^{9} + \)\(10\!\cdots\!66\)\( p^{129} T^{10} - \)\(38\!\cdots\!60\)\( p^{248} T^{11} + \)\(25\!\cdots\!52\)\( p^{367} T^{12} + \)\(94\!\cdots\!36\)\( p^{487} T^{13} + \)\(21\!\cdots\!84\)\( p^{607} T^{14} + \)\(17\!\cdots\!40\)\( p^{727} T^{15} + \)\(10\!\cdots\!07\)\( p^{847} T^{16} + \)\(13\!\cdots\!48\)\( p^{968} T^{17} + p^{1089} T^{18} \)
53 \( 1 - \)\(10\!\cdots\!46\)\( T + \)\(42\!\cdots\!21\)\( p T^{2} - \)\(10\!\cdots\!40\)\( p^{2} T^{3} + \)\(16\!\cdots\!08\)\( p^{3} T^{4} - \)\(49\!\cdots\!76\)\( p^{4} T^{5} + \)\(76\!\cdots\!32\)\( p^{6} T^{6} - \)\(55\!\cdots\!20\)\( p^{8} T^{7} + \)\(51\!\cdots\!54\)\( p^{10} T^{8} - \)\(39\!\cdots\!16\)\( p^{12} T^{9} + \)\(51\!\cdots\!54\)\( p^{131} T^{10} - \)\(55\!\cdots\!20\)\( p^{250} T^{11} + \)\(76\!\cdots\!32\)\( p^{369} T^{12} - \)\(49\!\cdots\!76\)\( p^{488} T^{13} + \)\(16\!\cdots\!08\)\( p^{608} T^{14} - \)\(10\!\cdots\!40\)\( p^{728} T^{15} + \)\(42\!\cdots\!21\)\( p^{848} T^{16} - \)\(10\!\cdots\!46\)\( p^{968} T^{17} + p^{1089} T^{18} \)
59 \( 1 + \)\(36\!\cdots\!40\)\( p T + \)\(29\!\cdots\!51\)\( p^{2} T^{2} + \)\(84\!\cdots\!80\)\( p^{3} T^{3} + \)\(42\!\cdots\!56\)\( p^{4} T^{4} + \)\(98\!\cdots\!20\)\( p^{5} T^{5} + \)\(64\!\cdots\!44\)\( p^{7} T^{6} + \)\(21\!\cdots\!60\)\( p^{9} T^{7} + \)\(12\!\cdots\!54\)\( p^{11} T^{8} + \)\(36\!\cdots\!00\)\( p^{13} T^{9} + \)\(12\!\cdots\!54\)\( p^{132} T^{10} + \)\(21\!\cdots\!60\)\( p^{251} T^{11} + \)\(64\!\cdots\!44\)\( p^{370} T^{12} + \)\(98\!\cdots\!20\)\( p^{489} T^{13} + \)\(42\!\cdots\!56\)\( p^{609} T^{14} + \)\(84\!\cdots\!80\)\( p^{729} T^{15} + \)\(29\!\cdots\!51\)\( p^{849} T^{16} + \)\(36\!\cdots\!40\)\( p^{969} T^{17} + p^{1089} T^{18} \)
61 \( 1 - \)\(11\!\cdots\!58\)\( T + \)\(75\!\cdots\!53\)\( p T^{2} - \)\(66\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(26\!\cdots\!88\)\( p T^{5} + \)\(54\!\cdots\!04\)\( p^{2} T^{6} - \)\(10\!\cdots\!44\)\( p^{3} T^{7} + \)\(20\!\cdots\!58\)\( p^{4} T^{8} - \)\(32\!\cdots\!20\)\( p^{5} T^{9} + \)\(20\!\cdots\!58\)\( p^{125} T^{10} - \)\(10\!\cdots\!44\)\( p^{245} T^{11} + \)\(54\!\cdots\!04\)\( p^{365} T^{12} - \)\(26\!\cdots\!88\)\( p^{485} T^{13} + \)\(11\!\cdots\!60\)\( p^{605} T^{14} - \)\(66\!\cdots\!56\)\( p^{726} T^{15} + \)\(75\!\cdots\!53\)\( p^{848} T^{16} - \)\(11\!\cdots\!58\)\( p^{968} T^{17} + p^{1089} T^{18} \)
67 \( 1 + \)\(13\!\cdots\!68\)\( T + \)\(32\!\cdots\!07\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!36\)\( T^{4} - \)\(54\!\cdots\!76\)\( p T^{5} + \)\(16\!\cdots\!88\)\( p^{2} T^{6} - \)\(17\!\cdots\!20\)\( p^{3} T^{7} + \)\(44\!\cdots\!66\)\( p^{4} T^{8} - \)\(34\!\cdots\!76\)\( p^{5} T^{9} + \)\(44\!\cdots\!66\)\( p^{125} T^{10} - \)\(17\!\cdots\!20\)\( p^{245} T^{11} + \)\(16\!\cdots\!88\)\( p^{365} T^{12} - \)\(54\!\cdots\!76\)\( p^{485} T^{13} + \)\(54\!\cdots\!36\)\( p^{605} T^{14} + \)\(26\!\cdots\!20\)\( p^{726} T^{15} + \)\(32\!\cdots\!07\)\( p^{847} T^{16} + \)\(13\!\cdots\!68\)\( p^{968} T^{17} + p^{1089} T^{18} \)
71 \( 1 + \)\(21\!\cdots\!52\)\( T + \)\(81\!\cdots\!63\)\( T^{2} + \)\(14\!\cdots\!04\)\( T^{3} + \)\(30\!\cdots\!60\)\( T^{4} + \)\(62\!\cdots\!92\)\( p T^{5} + \)\(13\!\cdots\!44\)\( p^{2} T^{6} + \)\(23\!\cdots\!36\)\( p^{3} T^{7} + \)\(40\!\cdots\!58\)\( p^{4} T^{8} + \)\(55\!\cdots\!80\)\( p^{5} T^{9} + \)\(40\!\cdots\!58\)\( p^{125} T^{10} + \)\(23\!\cdots\!36\)\( p^{245} T^{11} + \)\(13\!\cdots\!44\)\( p^{365} T^{12} + \)\(62\!\cdots\!92\)\( p^{485} T^{13} + \)\(30\!\cdots\!60\)\( p^{605} T^{14} + \)\(14\!\cdots\!04\)\( p^{726} T^{15} + \)\(81\!\cdots\!63\)\( p^{847} T^{16} + \)\(21\!\cdots\!52\)\( p^{968} T^{17} + p^{1089} T^{18} \)
73 \( 1 - \)\(91\!\cdots\!26\)\( T + \)\(16\!\cdots\!53\)\( T^{2} - \)\(12\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!72\)\( p T^{4} - \)\(17\!\cdots\!04\)\( p^{2} T^{5} + \)\(19\!\cdots\!84\)\( p^{3} T^{6} - \)\(15\!\cdots\!60\)\( p^{4} T^{7} + \)\(14\!\cdots\!82\)\( p^{5} T^{8} - \)\(10\!\cdots\!64\)\( p^{6} T^{9} + \)\(14\!\cdots\!82\)\( p^{126} T^{10} - \)\(15\!\cdots\!60\)\( p^{246} T^{11} + \)\(19\!\cdots\!84\)\( p^{366} T^{12} - \)\(17\!\cdots\!04\)\( p^{486} T^{13} + \)\(18\!\cdots\!72\)\( p^{606} T^{14} - \)\(12\!\cdots\!80\)\( p^{726} T^{15} + \)\(16\!\cdots\!53\)\( p^{847} T^{16} - \)\(91\!\cdots\!26\)\( p^{968} T^{17} + p^{1089} T^{18} \)
79 \( 1 - \)\(12\!\cdots\!20\)\( T + \)\(26\!\cdots\!11\)\( T^{2} - \)\(26\!\cdots\!60\)\( p T^{3} + \)\(36\!\cdots\!44\)\( p T^{4} - \)\(28\!\cdots\!60\)\( p^{2} T^{5} + \)\(40\!\cdots\!84\)\( p^{3} T^{6} - \)\(27\!\cdots\!80\)\( p^{4} T^{7} + \)\(34\!\cdots\!94\)\( p^{5} T^{8} - \)\(21\!\cdots\!00\)\( p^{6} T^{9} + \)\(34\!\cdots\!94\)\( p^{126} T^{10} - \)\(27\!\cdots\!80\)\( p^{246} T^{11} + \)\(40\!\cdots\!84\)\( p^{366} T^{12} - \)\(28\!\cdots\!60\)\( p^{486} T^{13} + \)\(36\!\cdots\!44\)\( p^{606} T^{14} - \)\(26\!\cdots\!60\)\( p^{727} T^{15} + \)\(26\!\cdots\!11\)\( p^{847} T^{16} - \)\(12\!\cdots\!20\)\( p^{968} T^{17} + p^{1089} T^{18} \)
83 \( 1 + \)\(92\!\cdots\!84\)\( T + \)\(79\!\cdots\!23\)\( T^{2} + \)\(51\!\cdots\!60\)\( T^{3} + \)\(38\!\cdots\!72\)\( p T^{4} + \)\(22\!\cdots\!36\)\( p^{2} T^{5} + \)\(15\!\cdots\!44\)\( p^{3} T^{6} + \)\(65\!\cdots\!20\)\( p^{4} T^{7} + \)\(45\!\cdots\!62\)\( p^{5} T^{8} + \)\(16\!\cdots\!36\)\( p^{6} T^{9} + \)\(45\!\cdots\!62\)\( p^{126} T^{10} + \)\(65\!\cdots\!20\)\( p^{246} T^{11} + \)\(15\!\cdots\!44\)\( p^{366} T^{12} + \)\(22\!\cdots\!36\)\( p^{486} T^{13} + \)\(38\!\cdots\!72\)\( p^{606} T^{14} + \)\(51\!\cdots\!60\)\( p^{726} T^{15} + \)\(79\!\cdots\!23\)\( p^{847} T^{16} + \)\(92\!\cdots\!84\)\( p^{968} T^{17} + p^{1089} T^{18} \)
89 \( 1 + \)\(60\!\cdots\!90\)\( T + \)\(32\!\cdots\!01\)\( T^{2} + \)\(14\!\cdots\!20\)\( p T^{3} + \)\(55\!\cdots\!36\)\( p^{2} T^{4} + \)\(21\!\cdots\!80\)\( p^{3} T^{5} + \)\(44\!\cdots\!56\)\( p^{4} T^{6} - \)\(38\!\cdots\!60\)\( p^{5} T^{7} + \)\(17\!\cdots\!06\)\( p^{6} T^{8} - \)\(61\!\cdots\!00\)\( p^{7} T^{9} + \)\(17\!\cdots\!06\)\( p^{127} T^{10} - \)\(38\!\cdots\!60\)\( p^{247} T^{11} + \)\(44\!\cdots\!56\)\( p^{367} T^{12} + \)\(21\!\cdots\!80\)\( p^{487} T^{13} + \)\(55\!\cdots\!36\)\( p^{607} T^{14} + \)\(14\!\cdots\!20\)\( p^{727} T^{15} + \)\(32\!\cdots\!01\)\( p^{847} T^{16} + \)\(60\!\cdots\!90\)\( p^{968} T^{17} + p^{1089} T^{18} \)
97 \( 1 + \)\(10\!\cdots\!98\)\( T + \)\(65\!\cdots\!81\)\( p T^{2} + \)\(30\!\cdots\!20\)\( p^{2} T^{3} + \)\(11\!\cdots\!72\)\( p^{3} T^{4} + \)\(33\!\cdots\!88\)\( p^{4} T^{5} + \)\(86\!\cdots\!16\)\( p^{5} T^{6} + \)\(19\!\cdots\!40\)\( p^{6} T^{7} + \)\(38\!\cdots\!02\)\( p^{7} T^{8} + \)\(66\!\cdots\!48\)\( p^{8} T^{9} + \)\(38\!\cdots\!02\)\( p^{128} T^{10} + \)\(19\!\cdots\!40\)\( p^{248} T^{11} + \)\(86\!\cdots\!16\)\( p^{368} T^{12} + \)\(33\!\cdots\!88\)\( p^{488} T^{13} + \)\(11\!\cdots\!72\)\( p^{608} T^{14} + \)\(30\!\cdots\!20\)\( p^{728} T^{15} + \)\(65\!\cdots\!81\)\( p^{848} T^{16} + \)\(10\!\cdots\!98\)\( p^{968} T^{17} + p^{1089} T^{18} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{18} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.42778080146646461605203938370, −4.29085246711873857986566204994, −4.14065537164408417517156886341, −3.98698517495385285832923279003, −3.87791261839878826557150969747, −3.86002858985582348197576897857, −3.67180130888566367765870692619, −3.37168855217321009935334229082, −3.17750159138648833008179514365, −3.09857052344960047162551893851, −3.07326211051504509230320216482, −2.80569422698728855637360791951, −2.55443015128312689108270457656, −2.55081171416203299290435805325, −2.02206111376412570107210411601, −2.01111300048748229802812515865, −1.98784233832501905113234329063, −1.84761196756697915752031374201, −1.57090422649161902579128446104, −1.54655743489228463980833212194, −1.33175679004398175827482061837, −1.04302590719381750002896100873, −1.03367236365577053618719383018, −0.796403657589990784788379447132, −0.74371437155340444799141446063, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.74371437155340444799141446063, 0.796403657589990784788379447132, 1.03367236365577053618719383018, 1.04302590719381750002896100873, 1.33175679004398175827482061837, 1.54655743489228463980833212194, 1.57090422649161902579128446104, 1.84761196756697915752031374201, 1.98784233832501905113234329063, 2.01111300048748229802812515865, 2.02206111376412570107210411601, 2.55081171416203299290435805325, 2.55443015128312689108270457656, 2.80569422698728855637360791951, 3.07326211051504509230320216482, 3.09857052344960047162551893851, 3.17750159138648833008179514365, 3.37168855217321009935334229082, 3.67180130888566367765870692619, 3.86002858985582348197576897857, 3.87791261839878826557150969747, 3.98698517495385285832923279003, 4.14065537164408417517156886341, 4.29085246711873857986566204994, 4.42778080146646461605203938370

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

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