Properties

Label 14-1-1.1-c87e7-0-0
Degree $14$
Conductor $1$
Sign $1$
Analytic cond. $5.81386\times 10^{11}$
Root an. cond. $6.92339$
Motivic weight $87$
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  + 1.81e13·2-s − 7.54e19·3-s − 1.87e26·4-s + 3.30e29·5-s − 1.37e33·6-s + 4.55e33·7-s − 5.43e39·8-s − 7.91e41·9-s + 6.01e42·10-s + 1.47e45·11-s + 1.41e46·12-s + 3.01e48·13-s + 8.29e46·14-s − 2.49e49·15-s − 1.00e52·16-s + 3.05e53·17-s − 1.44e55·18-s − 2.39e55·19-s − 6.20e55·20-s − 3.43e53·21-s + 2.67e58·22-s − 1.91e58·23-s + 4.09e59·24-s − 1.76e61·25-s + 5.48e61·26-s + 5.72e61·27-s − 8.56e59·28-s + ⋯
L(s)  = 1  + 1.46·2-s − 0.132·3-s − 1.21·4-s + 0.129·5-s − 0.194·6-s + 0.000789·7-s − 2.82·8-s − 2.44·9-s + 0.190·10-s + 0.736·11-s + 0.161·12-s + 1.05·13-s + 0.00115·14-s − 0.0172·15-s − 0.418·16-s + 0.912·17-s − 3.58·18-s − 0.565·19-s − 0.157·20-s − 0.000104·21-s + 1.07·22-s − 0.111·23-s + 0.374·24-s − 2.73·25-s + 1.54·26-s + 0.311·27-s − 0.000957·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(88-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+87/2)^{7} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(14\)
Conductor: \(1\)
Sign: $1$
Analytic conductor: \(5.81386\times 10^{11}\)
Root analytic conductor: \(6.92339\)
Motivic weight: \(87\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((14,\ 1,\ (\ :[87/2]^{7}),\ 1)\)

Particular Values

\(L(44)\) \(\approx\) \(1.521436044\)
\(L(\frac12)\) \(\approx\) \(1.521436044\)
\(L(\frac{89}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
good2 \( 1 - 2274627755367 p^{3} T + \)\(50\!\cdots\!47\)\( p^{10} T^{2} - \)\(35\!\cdots\!25\)\( p^{21} T^{3} + \)\(10\!\cdots\!41\)\( p^{37} T^{4} - \)\(12\!\cdots\!01\)\( p^{57} T^{5} + \)\(23\!\cdots\!31\)\( p^{80} T^{6} - \)\(19\!\cdots\!75\)\( p^{107} T^{7} + \)\(23\!\cdots\!31\)\( p^{167} T^{8} - \)\(12\!\cdots\!01\)\( p^{231} T^{9} + \)\(10\!\cdots\!41\)\( p^{298} T^{10} - \)\(35\!\cdots\!25\)\( p^{369} T^{11} + \)\(50\!\cdots\!47\)\( p^{445} T^{12} - 2274627755367 p^{525} T^{13} + p^{609} T^{14} \)
3 \( 1 + 2793229985119643524 p^{3} T + \)\(40\!\cdots\!39\)\( p^{9} T^{2} + \)\(16\!\cdots\!00\)\( p^{18} T^{3} + \)\(55\!\cdots\!31\)\( p^{31} T^{4} + \)\(33\!\cdots\!28\)\( p^{47} T^{5} + \)\(99\!\cdots\!67\)\( p^{63} T^{6} + \)\(32\!\cdots\!00\)\( p^{82} T^{7} + \)\(99\!\cdots\!67\)\( p^{150} T^{8} + \)\(33\!\cdots\!28\)\( p^{221} T^{9} + \)\(55\!\cdots\!31\)\( p^{292} T^{10} + \)\(16\!\cdots\!00\)\( p^{366} T^{11} + \)\(40\!\cdots\!39\)\( p^{444} T^{12} + 2793229985119643524 p^{525} T^{13} + p^{609} T^{14} \)
5 \( 1 - \)\(13\!\cdots\!54\)\( p^{2} T + \)\(56\!\cdots\!11\)\( p^{5} T^{2} - \)\(12\!\cdots\!32\)\( p^{9} T^{3} + \)\(10\!\cdots\!29\)\( p^{16} T^{4} - \)\(29\!\cdots\!14\)\( p^{26} T^{5} + \)\(29\!\cdots\!59\)\( p^{38} T^{6} - \)\(88\!\cdots\!96\)\( p^{51} T^{7} + \)\(29\!\cdots\!59\)\( p^{125} T^{8} - \)\(29\!\cdots\!14\)\( p^{200} T^{9} + \)\(10\!\cdots\!29\)\( p^{277} T^{10} - \)\(12\!\cdots\!32\)\( p^{357} T^{11} + \)\(56\!\cdots\!11\)\( p^{440} T^{12} - \)\(13\!\cdots\!54\)\( p^{524} T^{13} + p^{609} T^{14} \)
7 \( 1 - \)\(65\!\cdots\!08\)\( p T + \)\(18\!\cdots\!93\)\( p^{4} T^{2} - \)\(36\!\cdots\!00\)\( p^{8} T^{3} + \)\(32\!\cdots\!53\)\( p^{14} T^{4} + \)\(26\!\cdots\!24\)\( p^{21} T^{5} + \)\(27\!\cdots\!29\)\( p^{30} T^{6} + \)\(88\!\cdots\!00\)\( p^{40} T^{7} + \)\(27\!\cdots\!29\)\( p^{117} T^{8} + \)\(26\!\cdots\!24\)\( p^{195} T^{9} + \)\(32\!\cdots\!53\)\( p^{275} T^{10} - \)\(36\!\cdots\!00\)\( p^{356} T^{11} + \)\(18\!\cdots\!93\)\( p^{439} T^{12} - \)\(65\!\cdots\!08\)\( p^{523} T^{13} + p^{609} T^{14} \)
11 \( 1 - \)\(13\!\cdots\!64\)\( p T + \)\(14\!\cdots\!31\)\( p^{3} T^{2} - \)\(13\!\cdots\!44\)\( p^{6} T^{3} + \)\(78\!\cdots\!51\)\( p^{9} T^{4} - \)\(51\!\cdots\!92\)\( p^{13} T^{5} + \)\(19\!\cdots\!93\)\( p^{18} T^{6} - \)\(79\!\cdots\!72\)\( p^{25} T^{7} + \)\(19\!\cdots\!93\)\( p^{105} T^{8} - \)\(51\!\cdots\!92\)\( p^{187} T^{9} + \)\(78\!\cdots\!51\)\( p^{270} T^{10} - \)\(13\!\cdots\!44\)\( p^{354} T^{11} + \)\(14\!\cdots\!31\)\( p^{438} T^{12} - \)\(13\!\cdots\!64\)\( p^{523} T^{13} + p^{609} T^{14} \)
13 \( 1 - \)\(23\!\cdots\!74\)\( p T + \)\(15\!\cdots\!43\)\( p^{2} T^{2} - \)\(38\!\cdots\!00\)\( p^{3} T^{3} + \)\(92\!\cdots\!13\)\( p^{6} T^{4} - \)\(94\!\cdots\!46\)\( p^{10} T^{5} + \)\(99\!\cdots\!27\)\( p^{15} T^{6} - \)\(50\!\cdots\!00\)\( p^{21} T^{7} + \)\(99\!\cdots\!27\)\( p^{102} T^{8} - \)\(94\!\cdots\!46\)\( p^{184} T^{9} + \)\(92\!\cdots\!13\)\( p^{267} T^{10} - \)\(38\!\cdots\!00\)\( p^{351} T^{11} + \)\(15\!\cdots\!43\)\( p^{437} T^{12} - \)\(23\!\cdots\!74\)\( p^{523} T^{13} + p^{609} T^{14} \)
17 \( 1 - \)\(17\!\cdots\!38\)\( p T + \)\(10\!\cdots\!07\)\( p^{2} T^{2} - \)\(65\!\cdots\!00\)\( p^{4} T^{3} + \)\(14\!\cdots\!69\)\( p^{7} T^{4} - \)\(57\!\cdots\!98\)\( p^{10} T^{5} + \)\(86\!\cdots\!23\)\( p^{13} T^{6} - \)\(24\!\cdots\!00\)\( p^{16} T^{7} + \)\(86\!\cdots\!23\)\( p^{100} T^{8} - \)\(57\!\cdots\!98\)\( p^{184} T^{9} + \)\(14\!\cdots\!69\)\( p^{268} T^{10} - \)\(65\!\cdots\!00\)\( p^{352} T^{11} + \)\(10\!\cdots\!07\)\( p^{437} T^{12} - \)\(17\!\cdots\!38\)\( p^{523} T^{13} + p^{609} T^{14} \)
19 \( 1 + \)\(23\!\cdots\!40\)\( T + \)\(49\!\cdots\!67\)\( p T^{2} + \)\(24\!\cdots\!40\)\( p^{3} T^{3} + \)\(16\!\cdots\!59\)\( p^{5} T^{4} + \)\(32\!\cdots\!00\)\( p^{8} T^{5} + \)\(49\!\cdots\!65\)\( p^{12} T^{6} + \)\(40\!\cdots\!00\)\( p^{16} T^{7} + \)\(49\!\cdots\!65\)\( p^{99} T^{8} + \)\(32\!\cdots\!00\)\( p^{182} T^{9} + \)\(16\!\cdots\!59\)\( p^{266} T^{10} + \)\(24\!\cdots\!40\)\( p^{351} T^{11} + \)\(49\!\cdots\!67\)\( p^{436} T^{12} + \)\(23\!\cdots\!40\)\( p^{522} T^{13} + p^{609} T^{14} \)
23 \( 1 + \)\(19\!\cdots\!28\)\( T + \)\(50\!\cdots\!39\)\( p T^{2} - \)\(68\!\cdots\!00\)\( p^{2} T^{3} + \)\(23\!\cdots\!97\)\( p^{4} T^{4} - \)\(25\!\cdots\!96\)\( p^{6} T^{5} + \)\(14\!\cdots\!63\)\( p^{9} T^{6} - \)\(69\!\cdots\!00\)\( p^{12} T^{7} + \)\(14\!\cdots\!63\)\( p^{96} T^{8} - \)\(25\!\cdots\!96\)\( p^{180} T^{9} + \)\(23\!\cdots\!97\)\( p^{265} T^{10} - \)\(68\!\cdots\!00\)\( p^{350} T^{11} + \)\(50\!\cdots\!39\)\( p^{436} T^{12} + \)\(19\!\cdots\!28\)\( p^{522} T^{13} + p^{609} T^{14} \)
29 \( 1 + \)\(10\!\cdots\!90\)\( p T + \)\(11\!\cdots\!43\)\( p^{2} T^{2} + \)\(97\!\cdots\!60\)\( p^{3} T^{3} + \)\(60\!\cdots\!21\)\( p^{4} T^{4} + \)\(42\!\cdots\!50\)\( p^{5} T^{5} + \)\(18\!\cdots\!15\)\( p^{6} T^{6} + \)\(10\!\cdots\!00\)\( p^{7} T^{7} + \)\(18\!\cdots\!15\)\( p^{93} T^{8} + \)\(42\!\cdots\!50\)\( p^{179} T^{9} + \)\(60\!\cdots\!21\)\( p^{265} T^{10} + \)\(97\!\cdots\!60\)\( p^{351} T^{11} + \)\(11\!\cdots\!43\)\( p^{437} T^{12} + \)\(10\!\cdots\!90\)\( p^{523} T^{13} + p^{609} T^{14} \)
31 \( 1 - \)\(64\!\cdots\!84\)\( T + \)\(12\!\cdots\!01\)\( T^{2} + \)\(94\!\cdots\!96\)\( p T^{3} + \)\(92\!\cdots\!61\)\( p^{2} T^{4} + \)\(47\!\cdots\!88\)\( p^{4} T^{5} + \)\(48\!\cdots\!53\)\( p^{6} T^{6} + \)\(44\!\cdots\!28\)\( p^{8} T^{7} + \)\(48\!\cdots\!53\)\( p^{93} T^{8} + \)\(47\!\cdots\!88\)\( p^{178} T^{9} + \)\(92\!\cdots\!61\)\( p^{263} T^{10} + \)\(94\!\cdots\!96\)\( p^{349} T^{11} + \)\(12\!\cdots\!01\)\( p^{435} T^{12} - \)\(64\!\cdots\!84\)\( p^{522} T^{13} + p^{609} T^{14} \)
37 \( 1 + \)\(25\!\cdots\!74\)\( T + \)\(33\!\cdots\!59\)\( p T^{2} + \)\(16\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!89\)\( p^{3} T^{4} + \)\(10\!\cdots\!94\)\( p^{5} T^{5} + \)\(13\!\cdots\!67\)\( p^{7} T^{6} + \)\(12\!\cdots\!00\)\( p^{9} T^{7} + \)\(13\!\cdots\!67\)\( p^{94} T^{8} + \)\(10\!\cdots\!94\)\( p^{179} T^{9} + \)\(10\!\cdots\!89\)\( p^{264} T^{10} + \)\(16\!\cdots\!00\)\( p^{350} T^{11} + \)\(33\!\cdots\!59\)\( p^{436} T^{12} + \)\(25\!\cdots\!74\)\( p^{522} T^{13} + p^{609} T^{14} \)
41 \( 1 - \)\(49\!\cdots\!14\)\( p T + \)\(49\!\cdots\!91\)\( p^{2} T^{2} - \)\(47\!\cdots\!64\)\( p^{3} T^{3} + \)\(12\!\cdots\!01\)\( p^{4} T^{4} - \)\(37\!\cdots\!22\)\( p^{6} T^{5} + \)\(12\!\cdots\!63\)\( p^{8} T^{6} - \)\(37\!\cdots\!92\)\( p^{10} T^{7} + \)\(12\!\cdots\!63\)\( p^{95} T^{8} - \)\(37\!\cdots\!22\)\( p^{180} T^{9} + \)\(12\!\cdots\!01\)\( p^{265} T^{10} - \)\(47\!\cdots\!64\)\( p^{351} T^{11} + \)\(49\!\cdots\!91\)\( p^{437} T^{12} - \)\(49\!\cdots\!14\)\( p^{523} T^{13} + p^{609} T^{14} \)
43 \( 1 - \)\(72\!\cdots\!44\)\( p T + \)\(57\!\cdots\!93\)\( p^{2} T^{2} - \)\(26\!\cdots\!00\)\( p^{3} T^{3} + \)\(12\!\cdots\!97\)\( p^{4} T^{4} - \)\(94\!\cdots\!76\)\( p^{6} T^{5} + \)\(13\!\cdots\!21\)\( p^{6} T^{6} - \)\(36\!\cdots\!00\)\( p^{7} T^{7} + \)\(13\!\cdots\!21\)\( p^{93} T^{8} - \)\(94\!\cdots\!76\)\( p^{180} T^{9} + \)\(12\!\cdots\!97\)\( p^{265} T^{10} - \)\(26\!\cdots\!00\)\( p^{351} T^{11} + \)\(57\!\cdots\!93\)\( p^{437} T^{12} - \)\(72\!\cdots\!44\)\( p^{523} T^{13} + p^{609} T^{14} \)
47 \( 1 - \)\(11\!\cdots\!16\)\( T + \)\(15\!\cdots\!13\)\( T^{2} - \)\(10\!\cdots\!00\)\( T^{3} + \)\(20\!\cdots\!31\)\( p T^{4} - \)\(23\!\cdots\!68\)\( p^{2} T^{5} + \)\(35\!\cdots\!67\)\( p^{3} T^{6} - \)\(35\!\cdots\!00\)\( p^{4} T^{7} + \)\(35\!\cdots\!67\)\( p^{90} T^{8} - \)\(23\!\cdots\!68\)\( p^{176} T^{9} + \)\(20\!\cdots\!31\)\( p^{262} T^{10} - \)\(10\!\cdots\!00\)\( p^{348} T^{11} + \)\(15\!\cdots\!13\)\( p^{435} T^{12} - \)\(11\!\cdots\!16\)\( p^{522} T^{13} + p^{609} T^{14} \)
53 \( 1 - \)\(34\!\cdots\!02\)\( T + \)\(73\!\cdots\!87\)\( T^{2} - \)\(98\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!69\)\( p T^{4} - \)\(14\!\cdots\!46\)\( p^{2} T^{5} - \)\(10\!\cdots\!33\)\( p^{3} T^{6} + \)\(53\!\cdots\!00\)\( p^{4} T^{7} - \)\(10\!\cdots\!33\)\( p^{90} T^{8} - \)\(14\!\cdots\!46\)\( p^{176} T^{9} + \)\(16\!\cdots\!69\)\( p^{262} T^{10} - \)\(98\!\cdots\!00\)\( p^{348} T^{11} + \)\(73\!\cdots\!87\)\( p^{435} T^{12} - \)\(34\!\cdots\!02\)\( p^{522} T^{13} + p^{609} T^{14} \)
59 \( 1 + \)\(20\!\cdots\!20\)\( T + \)\(77\!\cdots\!33\)\( T^{2} + \)\(20\!\cdots\!20\)\( p T^{3} + \)\(71\!\cdots\!01\)\( p^{2} T^{4} + \)\(14\!\cdots\!00\)\( p^{3} T^{5} + \)\(63\!\cdots\!35\)\( p^{5} T^{6} + \)\(62\!\cdots\!00\)\( p^{5} T^{7} + \)\(63\!\cdots\!35\)\( p^{92} T^{8} + \)\(14\!\cdots\!00\)\( p^{177} T^{9} + \)\(71\!\cdots\!01\)\( p^{263} T^{10} + \)\(20\!\cdots\!20\)\( p^{349} T^{11} + \)\(77\!\cdots\!33\)\( p^{435} T^{12} + \)\(20\!\cdots\!20\)\( p^{522} T^{13} + p^{609} T^{14} \)
61 \( 1 + \)\(17\!\cdots\!46\)\( T + \)\(21\!\cdots\!11\)\( T^{2} + \)\(31\!\cdots\!56\)\( p T^{3} + \)\(38\!\cdots\!21\)\( p^{2} T^{4} + \)\(65\!\cdots\!78\)\( p^{4} T^{5} + \)\(59\!\cdots\!83\)\( p^{5} T^{6} + \)\(28\!\cdots\!28\)\( p^{5} T^{7} + \)\(59\!\cdots\!83\)\( p^{92} T^{8} + \)\(65\!\cdots\!78\)\( p^{178} T^{9} + \)\(38\!\cdots\!21\)\( p^{263} T^{10} + \)\(31\!\cdots\!56\)\( p^{349} T^{11} + \)\(21\!\cdots\!11\)\( p^{435} T^{12} + \)\(17\!\cdots\!46\)\( p^{522} T^{13} + p^{609} T^{14} \)
67 \( 1 - \)\(11\!\cdots\!96\)\( T + \)\(33\!\cdots\!19\)\( p T^{2} - \)\(14\!\cdots\!00\)\( p^{2} T^{3} + \)\(10\!\cdots\!99\)\( p^{3} T^{4} - \)\(46\!\cdots\!12\)\( p^{4} T^{5} + \)\(27\!\cdots\!43\)\( p^{5} T^{6} - \)\(82\!\cdots\!00\)\( p^{6} T^{7} + \)\(27\!\cdots\!43\)\( p^{92} T^{8} - \)\(46\!\cdots\!12\)\( p^{178} T^{9} + \)\(10\!\cdots\!99\)\( p^{264} T^{10} - \)\(14\!\cdots\!00\)\( p^{350} T^{11} + \)\(33\!\cdots\!19\)\( p^{436} T^{12} - \)\(11\!\cdots\!96\)\( p^{522} T^{13} + p^{609} T^{14} \)
71 \( 1 - \)\(11\!\cdots\!44\)\( T + \)\(13\!\cdots\!11\)\( p T^{2} - \)\(12\!\cdots\!44\)\( p^{2} T^{3} + \)\(90\!\cdots\!71\)\( p^{3} T^{4} - \)\(57\!\cdots\!92\)\( p^{4} T^{5} + \)\(32\!\cdots\!63\)\( p^{5} T^{6} - \)\(15\!\cdots\!72\)\( p^{6} T^{7} + \)\(32\!\cdots\!63\)\( p^{92} T^{8} - \)\(57\!\cdots\!92\)\( p^{178} T^{9} + \)\(90\!\cdots\!71\)\( p^{264} T^{10} - \)\(12\!\cdots\!44\)\( p^{350} T^{11} + \)\(13\!\cdots\!11\)\( p^{436} T^{12} - \)\(11\!\cdots\!44\)\( p^{522} T^{13} + p^{609} T^{14} \)
73 \( 1 + \)\(37\!\cdots\!78\)\( T + \)\(67\!\cdots\!39\)\( p T^{2} + \)\(19\!\cdots\!00\)\( p^{2} T^{3} + \)\(28\!\cdots\!81\)\( p^{3} T^{4} + \)\(75\!\cdots\!66\)\( p^{4} T^{5} + \)\(84\!\cdots\!83\)\( p^{5} T^{6} + \)\(24\!\cdots\!00\)\( p^{6} T^{7} + \)\(84\!\cdots\!83\)\( p^{92} T^{8} + \)\(75\!\cdots\!66\)\( p^{178} T^{9} + \)\(28\!\cdots\!81\)\( p^{264} T^{10} + \)\(19\!\cdots\!00\)\( p^{350} T^{11} + \)\(67\!\cdots\!39\)\( p^{436} T^{12} + \)\(37\!\cdots\!78\)\( p^{522} T^{13} + p^{609} T^{14} \)
79 \( 1 + \)\(10\!\cdots\!40\)\( p T + \)\(16\!\cdots\!93\)\( p^{2} T^{2} + \)\(12\!\cdots\!60\)\( p^{3} T^{3} + \)\(11\!\cdots\!21\)\( p^{4} T^{4} + \)\(62\!\cdots\!00\)\( p^{5} T^{5} + \)\(38\!\cdots\!65\)\( p^{6} T^{6} + \)\(16\!\cdots\!00\)\( p^{7} T^{7} + \)\(38\!\cdots\!65\)\( p^{93} T^{8} + \)\(62\!\cdots\!00\)\( p^{179} T^{9} + \)\(11\!\cdots\!21\)\( p^{265} T^{10} + \)\(12\!\cdots\!60\)\( p^{351} T^{11} + \)\(16\!\cdots\!93\)\( p^{437} T^{12} + \)\(10\!\cdots\!40\)\( p^{523} T^{13} + p^{609} T^{14} \)
83 \( 1 - \)\(96\!\cdots\!04\)\( p T + \)\(10\!\cdots\!93\)\( p^{2} T^{2} - \)\(64\!\cdots\!00\)\( p^{3} T^{3} + \)\(48\!\cdots\!59\)\( p^{5} T^{4} - \)\(19\!\cdots\!88\)\( p^{5} T^{5} + \)\(87\!\cdots\!21\)\( p^{6} T^{6} - \)\(32\!\cdots\!00\)\( p^{7} T^{7} + \)\(87\!\cdots\!21\)\( p^{93} T^{8} - \)\(19\!\cdots\!88\)\( p^{179} T^{9} + \)\(48\!\cdots\!59\)\( p^{266} T^{10} - \)\(64\!\cdots\!00\)\( p^{351} T^{11} + \)\(10\!\cdots\!93\)\( p^{437} T^{12} - \)\(96\!\cdots\!04\)\( p^{523} T^{13} + p^{609} T^{14} \)
89 \( 1 + \)\(21\!\cdots\!30\)\( T + \)\(36\!\cdots\!03\)\( T^{2} + \)\(42\!\cdots\!20\)\( T^{3} + \)\(44\!\cdots\!61\)\( T^{4} + \)\(37\!\cdots\!50\)\( T^{5} + \)\(28\!\cdots\!15\)\( T^{6} + \)\(19\!\cdots\!00\)\( T^{7} + \)\(28\!\cdots\!15\)\( p^{87} T^{8} + \)\(37\!\cdots\!50\)\( p^{174} T^{9} + \)\(44\!\cdots\!61\)\( p^{261} T^{10} + \)\(42\!\cdots\!20\)\( p^{348} T^{11} + \)\(36\!\cdots\!03\)\( p^{435} T^{12} + \)\(21\!\cdots\!30\)\( p^{522} T^{13} + p^{609} T^{14} \)
97 \( 1 - \)\(60\!\cdots\!66\)\( T + \)\(41\!\cdots\!63\)\( T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(57\!\cdots\!57\)\( T^{4} - \)\(15\!\cdots\!62\)\( T^{5} + \)\(42\!\cdots\!91\)\( T^{6} - \)\(10\!\cdots\!00\)\( T^{7} + \)\(42\!\cdots\!91\)\( p^{87} T^{8} - \)\(15\!\cdots\!62\)\( p^{174} T^{9} + \)\(57\!\cdots\!57\)\( p^{261} T^{10} - \)\(15\!\cdots\!00\)\( p^{348} T^{11} + \)\(41\!\cdots\!63\)\( p^{435} T^{12} - \)\(60\!\cdots\!66\)\( p^{522} T^{13} + p^{609} T^{14} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−5.61541317202669718833250014045, −5.55722976505826817465571579318, −5.20984251773182405089859809289, −5.06655878332264034460212201704, −4.48762186852133638366691883999, −4.47486283822637940380563910286, −4.46387001692769218690422696867, −3.95101594194089168369779118674, −3.87051340394422213480204515024, −3.71021238724494027693721385244, −3.65347321122838347249598890075, −3.22021912053991559863468258748, −3.05710998724344535873381840909, −2.75096612516943840057965160499, −2.55258974218499163699045597480, −2.32033065717603016412095847727, −1.99766475071213553700683468274, −1.69019304102728573214608690338, −1.66669921752842481514526654069, −1.27101621026077824899442848284, −0.946401193654180600381204881757, −0.68104828465265253401117786909, −0.44451601476954305112527875290, −0.38736706094937514541146427892, −0.11796714819926213086850200055, 0.11796714819926213086850200055, 0.38736706094937514541146427892, 0.44451601476954305112527875290, 0.68104828465265253401117786909, 0.946401193654180600381204881757, 1.27101621026077824899442848284, 1.66669921752842481514526654069, 1.69019304102728573214608690338, 1.99766475071213553700683468274, 2.32033065717603016412095847727, 2.55258974218499163699045597480, 2.75096612516943840057965160499, 3.05710998724344535873381840909, 3.22021912053991559863468258748, 3.65347321122838347249598890075, 3.71021238724494027693721385244, 3.87051340394422213480204515024, 3.95101594194089168369779118674, 4.46387001692769218690422696867, 4.47486283822637940380563910286, 4.48762186852133638366691883999, 5.06655878332264034460212201704, 5.20984251773182405089859809289, 5.55722976505826817465571579318, 5.61541317202669718833250014045

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

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