Properties

Degree 14
Conductor $ 1 $
Sign $-1$
Motivic weight 89
Primitive no
Self-dual yes
Analytic rank 7

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.14e13·2-s − 1.35e21·3-s − 5.42e26·4-s + 1.02e31·5-s + 4.27e34·6-s + 3.85e37·7-s + 2.78e40·8-s − 6.43e42·9-s − 3.21e44·10-s − 3.31e46·11-s + 7.37e47·12-s − 1.03e50·13-s − 1.20e51·14-s − 1.39e52·15-s − 8.86e51·16-s + 8.31e54·17-s + 2.02e56·18-s + 5.66e56·19-s − 5.54e57·20-s − 5.23e58·21-s + 1.04e60·22-s − 1.16e60·23-s − 3.78e61·24-s − 6.19e62·25-s + 3.24e63·26-s + 1.46e64·27-s − 2.08e64·28-s + ⋯
L(s)  = 1  − 1.26·2-s − 0.797·3-s − 0.876·4-s + 0.805·5-s + 1.00·6-s + 0.951·7-s + 1.80·8-s − 2.21·9-s − 1.01·10-s − 1.50·11-s + 0.698·12-s − 2.77·13-s − 1.20·14-s − 0.641·15-s − 0.0231·16-s + 1.46·17-s + 2.79·18-s + 0.705·19-s − 0.705·20-s − 0.758·21-s + 1.90·22-s − 0.294·23-s − 1.44·24-s − 3.83·25-s + 3.50·26-s + 2.95·27-s − 0.834·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(90-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+89/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(89\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(7\)
Selberg data  =  \((14,\ 1,\ (\ :[89/2]^{7}),\ -1)\)
\(L(45)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{91}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where,\(F_p(T)\) is a polynomial of degree 14.
$p$$F_p(T)$
good2 \( 1 + 1962958146963 p^{4} T + \)\(46\!\cdots\!29\)\( p^{15} T^{2} + \)\(27\!\cdots\!35\)\( p^{27} T^{3} + \)\(10\!\cdots\!27\)\( p^{40} T^{4} + \)\(10\!\cdots\!03\)\( p^{61} T^{5} + \)\(63\!\cdots\!51\)\( p^{86} T^{6} + \)\(57\!\cdots\!35\)\( p^{114} T^{7} + \)\(63\!\cdots\!51\)\( p^{175} T^{8} + \)\(10\!\cdots\!03\)\( p^{239} T^{9} + \)\(10\!\cdots\!27\)\( p^{307} T^{10} + \)\(27\!\cdots\!35\)\( p^{383} T^{11} + \)\(46\!\cdots\!29\)\( p^{460} T^{12} + 1962958146963 p^{538} T^{13} + p^{623} T^{14} \)
3 \( 1 + \)\(15\!\cdots\!96\)\( p^{2} T + \)\(12\!\cdots\!73\)\( p^{8} T^{2} + \)\(12\!\cdots\!80\)\( p^{16} T^{3} + \)\(25\!\cdots\!71\)\( p^{27} T^{4} - \)\(12\!\cdots\!56\)\( p^{43} T^{5} + \)\(15\!\cdots\!21\)\( p^{60} T^{6} - \)\(27\!\cdots\!40\)\( p^{78} T^{7} + \)\(15\!\cdots\!21\)\( p^{149} T^{8} - \)\(12\!\cdots\!56\)\( p^{221} T^{9} + \)\(25\!\cdots\!71\)\( p^{294} T^{10} + \)\(12\!\cdots\!80\)\( p^{372} T^{11} + \)\(12\!\cdots\!73\)\( p^{453} T^{12} + \)\(15\!\cdots\!96\)\( p^{536} T^{13} + p^{623} T^{14} \)
5 \( 1 - \)\(81\!\cdots\!58\)\( p^{3} T + \)\(92\!\cdots\!43\)\( p^{7} T^{2} - \)\(27\!\cdots\!88\)\( p^{12} T^{3} + \)\(50\!\cdots\!09\)\( p^{21} T^{4} - \)\(43\!\cdots\!22\)\( p^{31} T^{5} + \)\(89\!\cdots\!39\)\( p^{44} T^{6} - \)\(11\!\cdots\!64\)\( p^{58} T^{7} + \)\(89\!\cdots\!39\)\( p^{133} T^{8} - \)\(43\!\cdots\!22\)\( p^{209} T^{9} + \)\(50\!\cdots\!09\)\( p^{288} T^{10} - \)\(27\!\cdots\!88\)\( p^{368} T^{11} + \)\(92\!\cdots\!43\)\( p^{452} T^{12} - \)\(81\!\cdots\!58\)\( p^{537} T^{13} + p^{623} T^{14} \)
7 \( 1 - \)\(78\!\cdots\!08\)\( p^{2} T + \)\(33\!\cdots\!57\)\( p^{4} T^{2} - \)\(23\!\cdots\!00\)\( p^{6} T^{3} + \)\(15\!\cdots\!79\)\( p^{11} T^{4} - \)\(57\!\cdots\!76\)\( p^{18} T^{5} + \)\(81\!\cdots\!21\)\( p^{26} T^{6} - \)\(72\!\cdots\!00\)\( p^{36} T^{7} + \)\(81\!\cdots\!21\)\( p^{115} T^{8} - \)\(57\!\cdots\!76\)\( p^{196} T^{9} + \)\(15\!\cdots\!79\)\( p^{278} T^{10} - \)\(23\!\cdots\!00\)\( p^{362} T^{11} + \)\(33\!\cdots\!57\)\( p^{449} T^{12} - \)\(78\!\cdots\!08\)\( p^{536} T^{13} + p^{623} T^{14} \)
11 \( 1 + \)\(30\!\cdots\!96\)\( p T + \)\(13\!\cdots\!51\)\( p^{3} T^{2} + \)\(28\!\cdots\!36\)\( p^{6} T^{3} + \)\(74\!\cdots\!81\)\( p^{10} T^{4} + \)\(11\!\cdots\!28\)\( p^{14} T^{5} + \)\(21\!\cdots\!43\)\( p^{19} T^{6} + \)\(23\!\cdots\!88\)\( p^{25} T^{7} + \)\(21\!\cdots\!43\)\( p^{108} T^{8} + \)\(11\!\cdots\!28\)\( p^{192} T^{9} + \)\(74\!\cdots\!81\)\( p^{277} T^{10} + \)\(28\!\cdots\!36\)\( p^{362} T^{11} + \)\(13\!\cdots\!51\)\( p^{448} T^{12} + \)\(30\!\cdots\!96\)\( p^{535} T^{13} + p^{623} T^{14} \)
13 \( 1 + \)\(79\!\cdots\!18\)\( p T + \)\(36\!\cdots\!79\)\( p^{3} T^{2} + \)\(98\!\cdots\!20\)\( p^{5} T^{3} + \)\(20\!\cdots\!21\)\( p^{7} T^{4} + \)\(17\!\cdots\!42\)\( p^{10} T^{5} + \)\(27\!\cdots\!03\)\( p^{15} T^{6} - \)\(53\!\cdots\!40\)\( p^{21} T^{7} + \)\(27\!\cdots\!03\)\( p^{104} T^{8} + \)\(17\!\cdots\!42\)\( p^{188} T^{9} + \)\(20\!\cdots\!21\)\( p^{274} T^{10} + \)\(98\!\cdots\!20\)\( p^{361} T^{11} + \)\(36\!\cdots\!79\)\( p^{448} T^{12} + \)\(79\!\cdots\!18\)\( p^{535} T^{13} + p^{623} T^{14} \)
17 \( 1 - \)\(48\!\cdots\!26\)\( p T + \)\(32\!\cdots\!79\)\( p^{3} T^{2} - \)\(86\!\cdots\!80\)\( p^{5} T^{3} + \)\(17\!\cdots\!57\)\( p^{8} T^{4} - \)\(14\!\cdots\!94\)\( p^{12} T^{5} + \)\(12\!\cdots\!39\)\( p^{16} T^{6} - \)\(83\!\cdots\!80\)\( p^{20} T^{7} + \)\(12\!\cdots\!39\)\( p^{105} T^{8} - \)\(14\!\cdots\!94\)\( p^{190} T^{9} + \)\(17\!\cdots\!57\)\( p^{275} T^{10} - \)\(86\!\cdots\!80\)\( p^{361} T^{11} + \)\(32\!\cdots\!79\)\( p^{448} T^{12} - \)\(48\!\cdots\!26\)\( p^{535} T^{13} + p^{623} T^{14} \)
19 \( 1 - \)\(56\!\cdots\!80\)\( T + \)\(14\!\cdots\!87\)\( p T^{2} - \)\(44\!\cdots\!20\)\( p^{2} T^{3} + \)\(30\!\cdots\!41\)\( p^{4} T^{4} - \)\(23\!\cdots\!00\)\( p^{7} T^{5} + \)\(61\!\cdots\!65\)\( p^{10} T^{6} - \)\(20\!\cdots\!00\)\( p^{14} T^{7} + \)\(61\!\cdots\!65\)\( p^{99} T^{8} - \)\(23\!\cdots\!00\)\( p^{185} T^{9} + \)\(30\!\cdots\!41\)\( p^{271} T^{10} - \)\(44\!\cdots\!20\)\( p^{358} T^{11} + \)\(14\!\cdots\!87\)\( p^{446} T^{12} - \)\(56\!\cdots\!80\)\( p^{534} T^{13} + p^{623} T^{14} \)
23 \( 1 + \)\(11\!\cdots\!04\)\( T + \)\(63\!\cdots\!73\)\( T^{2} + \)\(17\!\cdots\!80\)\( p T^{3} + \)\(16\!\cdots\!11\)\( p^{3} T^{4} + \)\(77\!\cdots\!96\)\( p^{5} T^{5} + \)\(53\!\cdots\!41\)\( p^{8} T^{6} + \)\(40\!\cdots\!60\)\( p^{11} T^{7} + \)\(53\!\cdots\!41\)\( p^{97} T^{8} + \)\(77\!\cdots\!96\)\( p^{183} T^{9} + \)\(16\!\cdots\!11\)\( p^{270} T^{10} + \)\(17\!\cdots\!80\)\( p^{357} T^{11} + \)\(63\!\cdots\!73\)\( p^{445} T^{12} + \)\(11\!\cdots\!04\)\( p^{534} T^{13} + p^{623} T^{14} \)
29 \( 1 + \)\(48\!\cdots\!70\)\( p T + \)\(50\!\cdots\!63\)\( p^{2} T^{2} + \)\(56\!\cdots\!20\)\( p^{4} T^{3} + \)\(14\!\cdots\!61\)\( p^{6} T^{4} + \)\(10\!\cdots\!50\)\( p^{8} T^{5} + \)\(30\!\cdots\!15\)\( p^{10} T^{6} + \)\(20\!\cdots\!00\)\( p^{12} T^{7} + \)\(30\!\cdots\!15\)\( p^{99} T^{8} + \)\(10\!\cdots\!50\)\( p^{186} T^{9} + \)\(14\!\cdots\!61\)\( p^{273} T^{10} + \)\(56\!\cdots\!20\)\( p^{360} T^{11} + \)\(50\!\cdots\!63\)\( p^{447} T^{12} + \)\(48\!\cdots\!70\)\( p^{535} T^{13} + p^{623} T^{14} \)
31 \( 1 - \)\(68\!\cdots\!04\)\( T + \)\(21\!\cdots\!61\)\( T^{2} - \)\(73\!\cdots\!64\)\( p T^{3} + \)\(21\!\cdots\!81\)\( p^{2} T^{4} - \)\(10\!\cdots\!72\)\( p^{3} T^{5} + \)\(48\!\cdots\!83\)\( p^{5} T^{6} - \)\(77\!\cdots\!52\)\( p^{7} T^{7} + \)\(48\!\cdots\!83\)\( p^{94} T^{8} - \)\(10\!\cdots\!72\)\( p^{181} T^{9} + \)\(21\!\cdots\!81\)\( p^{269} T^{10} - \)\(73\!\cdots\!64\)\( p^{357} T^{11} + \)\(21\!\cdots\!61\)\( p^{445} T^{12} - \)\(68\!\cdots\!04\)\( p^{534} T^{13} + p^{623} T^{14} \)
37 \( 1 + \)\(54\!\cdots\!58\)\( T + \)\(46\!\cdots\!91\)\( p T^{2} + \)\(11\!\cdots\!80\)\( p^{2} T^{3} + \)\(28\!\cdots\!89\)\( p^{3} T^{4} + \)\(22\!\cdots\!78\)\( p^{5} T^{5} + \)\(82\!\cdots\!43\)\( p^{7} T^{6} + \)\(57\!\cdots\!80\)\( p^{9} T^{7} + \)\(82\!\cdots\!43\)\( p^{96} T^{8} + \)\(22\!\cdots\!78\)\( p^{183} T^{9} + \)\(28\!\cdots\!89\)\( p^{270} T^{10} + \)\(11\!\cdots\!80\)\( p^{358} T^{11} + \)\(46\!\cdots\!91\)\( p^{446} T^{12} + \)\(54\!\cdots\!58\)\( p^{534} T^{13} + p^{623} T^{14} \)
41 \( 1 + \)\(16\!\cdots\!26\)\( p T + \)\(88\!\cdots\!71\)\( p^{2} T^{2} + \)\(16\!\cdots\!56\)\( p^{3} T^{3} + \)\(10\!\cdots\!21\)\( p^{5} T^{4} + \)\(42\!\cdots\!58\)\( p^{7} T^{5} + \)\(19\!\cdots\!63\)\( p^{9} T^{6} + \)\(64\!\cdots\!28\)\( p^{11} T^{7} + \)\(19\!\cdots\!63\)\( p^{98} T^{8} + \)\(42\!\cdots\!58\)\( p^{185} T^{9} + \)\(10\!\cdots\!21\)\( p^{272} T^{10} + \)\(16\!\cdots\!56\)\( p^{359} T^{11} + \)\(88\!\cdots\!71\)\( p^{447} T^{12} + \)\(16\!\cdots\!26\)\( p^{535} T^{13} + p^{623} T^{14} \)
43 \( 1 - \)\(74\!\cdots\!92\)\( p T + \)\(54\!\cdots\!57\)\( p^{2} T^{2} - \)\(30\!\cdots\!00\)\( p^{3} T^{3} + \)\(14\!\cdots\!97\)\( p^{4} T^{4} - \)\(14\!\cdots\!68\)\( p^{6} T^{5} + \)\(13\!\cdots\!21\)\( p^{8} T^{6} - \)\(11\!\cdots\!00\)\( p^{10} T^{7} + \)\(13\!\cdots\!21\)\( p^{97} T^{8} - \)\(14\!\cdots\!68\)\( p^{184} T^{9} + \)\(14\!\cdots\!97\)\( p^{271} T^{10} - \)\(30\!\cdots\!00\)\( p^{359} T^{11} + \)\(54\!\cdots\!57\)\( p^{447} T^{12} - \)\(74\!\cdots\!92\)\( p^{535} T^{13} + p^{623} T^{14} \)
47 \( 1 + \)\(58\!\cdots\!08\)\( T + \)\(50\!\cdots\!37\)\( T^{2} + \)\(21\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!31\)\( p T^{4} + \)\(15\!\cdots\!04\)\( p^{2} T^{5} + \)\(11\!\cdots\!63\)\( p^{3} T^{6} + \)\(59\!\cdots\!80\)\( p^{4} T^{7} + \)\(11\!\cdots\!63\)\( p^{92} T^{8} + \)\(15\!\cdots\!04\)\( p^{180} T^{9} + \)\(22\!\cdots\!31\)\( p^{268} T^{10} + \)\(21\!\cdots\!60\)\( p^{356} T^{11} + \)\(50\!\cdots\!37\)\( p^{445} T^{12} + \)\(58\!\cdots\!08\)\( p^{534} T^{13} + p^{623} T^{14} \)
53 \( 1 + \)\(18\!\cdots\!14\)\( T + \)\(25\!\cdots\!03\)\( T^{2} + \)\(50\!\cdots\!60\)\( p T^{3} + \)\(83\!\cdots\!53\)\( p^{2} T^{4} + \)\(11\!\cdots\!94\)\( p^{3} T^{5} + \)\(14\!\cdots\!91\)\( p^{4} T^{6} + \)\(15\!\cdots\!80\)\( p^{5} T^{7} + \)\(14\!\cdots\!91\)\( p^{93} T^{8} + \)\(11\!\cdots\!94\)\( p^{181} T^{9} + \)\(83\!\cdots\!53\)\( p^{269} T^{10} + \)\(50\!\cdots\!60\)\( p^{357} T^{11} + \)\(25\!\cdots\!03\)\( p^{445} T^{12} + \)\(18\!\cdots\!14\)\( p^{534} T^{13} + p^{623} T^{14} \)
59 \( 1 + \)\(90\!\cdots\!60\)\( T + \)\(21\!\cdots\!73\)\( T^{2} + \)\(23\!\cdots\!60\)\( p T^{3} + \)\(56\!\cdots\!61\)\( p^{2} T^{4} + \)\(48\!\cdots\!00\)\( p^{3} T^{5} + \)\(92\!\cdots\!65\)\( p^{4} T^{6} + \)\(66\!\cdots\!00\)\( p^{5} T^{7} + \)\(92\!\cdots\!65\)\( p^{93} T^{8} + \)\(48\!\cdots\!00\)\( p^{181} T^{9} + \)\(56\!\cdots\!61\)\( p^{269} T^{10} + \)\(23\!\cdots\!60\)\( p^{357} T^{11} + \)\(21\!\cdots\!73\)\( p^{445} T^{12} + \)\(90\!\cdots\!60\)\( p^{534} T^{13} + p^{623} T^{14} \)
61 \( 1 - \)\(87\!\cdots\!94\)\( T + \)\(29\!\cdots\!31\)\( T^{2} - \)\(40\!\cdots\!64\)\( p T^{3} + \)\(11\!\cdots\!61\)\( p^{2} T^{4} - \)\(18\!\cdots\!42\)\( p^{3} T^{5} + \)\(33\!\cdots\!43\)\( p^{4} T^{6} - \)\(51\!\cdots\!12\)\( p^{5} T^{7} + \)\(33\!\cdots\!43\)\( p^{93} T^{8} - \)\(18\!\cdots\!42\)\( p^{181} T^{9} + \)\(11\!\cdots\!61\)\( p^{269} T^{10} - \)\(40\!\cdots\!64\)\( p^{357} T^{11} + \)\(29\!\cdots\!31\)\( p^{445} T^{12} - \)\(87\!\cdots\!94\)\( p^{534} T^{13} + p^{623} T^{14} \)
67 \( 1 - \)\(58\!\cdots\!92\)\( T + \)\(34\!\cdots\!77\)\( T^{2} - \)\(18\!\cdots\!80\)\( p T^{3} + \)\(90\!\cdots\!33\)\( p^{2} T^{4} - \)\(34\!\cdots\!68\)\( p^{3} T^{5} + \)\(12\!\cdots\!29\)\( p^{4} T^{6} - \)\(33\!\cdots\!40\)\( p^{5} T^{7} + \)\(12\!\cdots\!29\)\( p^{93} T^{8} - \)\(34\!\cdots\!68\)\( p^{181} T^{9} + \)\(90\!\cdots\!33\)\( p^{269} T^{10} - \)\(18\!\cdots\!80\)\( p^{357} T^{11} + \)\(34\!\cdots\!77\)\( p^{445} T^{12} - \)\(58\!\cdots\!92\)\( p^{534} T^{13} + p^{623} T^{14} \)
71 \( 1 + \)\(54\!\cdots\!76\)\( T + \)\(54\!\cdots\!51\)\( p T^{2} + \)\(29\!\cdots\!16\)\( p^{2} T^{3} + \)\(17\!\cdots\!51\)\( p^{3} T^{4} + \)\(72\!\cdots\!08\)\( p^{4} T^{5} + \)\(32\!\cdots\!23\)\( p^{5} T^{6} + \)\(10\!\cdots\!48\)\( p^{6} T^{7} + \)\(32\!\cdots\!23\)\( p^{94} T^{8} + \)\(72\!\cdots\!08\)\( p^{182} T^{9} + \)\(17\!\cdots\!51\)\( p^{270} T^{10} + \)\(29\!\cdots\!16\)\( p^{358} T^{11} + \)\(54\!\cdots\!51\)\( p^{446} T^{12} + \)\(54\!\cdots\!76\)\( p^{534} T^{13} + p^{623} T^{14} \)
73 \( 1 + \)\(19\!\cdots\!54\)\( T + \)\(73\!\cdots\!51\)\( p T^{2} + \)\(14\!\cdots\!60\)\( p^{2} T^{3} + \)\(30\!\cdots\!61\)\( p^{3} T^{4} + \)\(46\!\cdots\!58\)\( p^{4} T^{5} + \)\(66\!\cdots\!47\)\( p^{5} T^{6} + \)\(79\!\cdots\!80\)\( p^{6} T^{7} + \)\(66\!\cdots\!47\)\( p^{94} T^{8} + \)\(46\!\cdots\!58\)\( p^{182} T^{9} + \)\(30\!\cdots\!61\)\( p^{270} T^{10} + \)\(14\!\cdots\!60\)\( p^{358} T^{11} + \)\(73\!\cdots\!51\)\( p^{446} T^{12} + \)\(19\!\cdots\!54\)\( p^{534} T^{13} + p^{623} T^{14} \)
79 \( 1 - \)\(26\!\cdots\!20\)\( T + \)\(24\!\cdots\!27\)\( p T^{2} - \)\(35\!\cdots\!80\)\( p^{2} T^{3} + \)\(28\!\cdots\!79\)\( p^{3} T^{4} - \)\(40\!\cdots\!00\)\( p^{4} T^{5} + \)\(45\!\cdots\!35\)\( p^{5} T^{6} - \)\(10\!\cdots\!00\)\( p^{6} T^{7} + \)\(45\!\cdots\!35\)\( p^{94} T^{8} - \)\(40\!\cdots\!00\)\( p^{182} T^{9} + \)\(28\!\cdots\!79\)\( p^{270} T^{10} - \)\(35\!\cdots\!80\)\( p^{358} T^{11} + \)\(24\!\cdots\!27\)\( p^{446} T^{12} - \)\(26\!\cdots\!20\)\( p^{534} T^{13} + p^{623} T^{14} \)
83 \( 1 + \)\(42\!\cdots\!28\)\( p T + \)\(49\!\cdots\!97\)\( p^{2} T^{2} + \)\(14\!\cdots\!60\)\( p^{3} T^{3} + \)\(96\!\cdots\!77\)\( p^{4} T^{4} + \)\(19\!\cdots\!36\)\( p^{5} T^{5} + \)\(11\!\cdots\!09\)\( p^{6} T^{6} + \)\(19\!\cdots\!80\)\( p^{7} T^{7} + \)\(11\!\cdots\!09\)\( p^{95} T^{8} + \)\(19\!\cdots\!36\)\( p^{183} T^{9} + \)\(96\!\cdots\!77\)\( p^{271} T^{10} + \)\(14\!\cdots\!60\)\( p^{359} T^{11} + \)\(49\!\cdots\!97\)\( p^{447} T^{12} + \)\(42\!\cdots\!28\)\( p^{535} T^{13} + p^{623} T^{14} \)
89 \( 1 + \)\(16\!\cdots\!90\)\( T + \)\(25\!\cdots\!63\)\( T^{2} + \)\(22\!\cdots\!60\)\( T^{3} + \)\(20\!\cdots\!01\)\( T^{4} + \)\(13\!\cdots\!50\)\( T^{5} + \)\(88\!\cdots\!15\)\( T^{6} + \)\(47\!\cdots\!00\)\( T^{7} + \)\(88\!\cdots\!15\)\( p^{89} T^{8} + \)\(13\!\cdots\!50\)\( p^{178} T^{9} + \)\(20\!\cdots\!01\)\( p^{267} T^{10} + \)\(22\!\cdots\!60\)\( p^{356} T^{11} + \)\(25\!\cdots\!63\)\( p^{445} T^{12} + \)\(16\!\cdots\!90\)\( p^{534} T^{13} + p^{623} T^{14} \)
97 \( 1 - \)\(71\!\cdots\!42\)\( T + \)\(48\!\cdots\!87\)\( T^{2} - \)\(20\!\cdots\!40\)\( T^{3} + \)\(79\!\cdots\!57\)\( T^{4} - \)\(24\!\cdots\!14\)\( T^{5} + \)\(72\!\cdots\!99\)\( T^{6} - \)\(19\!\cdots\!20\)\( T^{7} + \)\(72\!\cdots\!99\)\( p^{89} T^{8} - \)\(24\!\cdots\!14\)\( p^{178} T^{9} + \)\(79\!\cdots\!57\)\( p^{267} T^{10} - \)\(20\!\cdots\!40\)\( p^{356} T^{11} + \)\(48\!\cdots\!87\)\( p^{445} T^{12} - \)\(71\!\cdots\!42\)\( p^{534} T^{13} + p^{623} T^{14} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−6.19462237036591940926645435560, −6.07621847382744094365184216419, −5.83807579188317149847073664613, −5.72401450927163662784671482604, −5.40996499257468715029139143785, −5.21415148062023115787017345883, −5.10416049686817180215745340848, −5.03128012492627329283061143280, −4.68605182707309679630770282207, −4.46793328136069584504190941734, −4.44123371708524621083614394267, −3.84868474186927879830908423001, −3.50641892894935739851514565752, −3.28763863480721116545096077530, −3.22121415116759283032440267067, −3.03532685951216845715275839283, −2.59148655381906762217474663683, −2.46657337848958541233786904618, −2.21905934986599851932169852405, −2.00434077516919755157598282934, −1.84015589183348535662397937312, −1.39266514465652854674679254581, −1.29402578060258513574725849796, −1.20485868071045966707975881821, −0.917234848080778442013681445998, 0, 0, 0, 0, 0, 0, 0, 0.917234848080778442013681445998, 1.20485868071045966707975881821, 1.29402578060258513574725849796, 1.39266514465652854674679254581, 1.84015589183348535662397937312, 2.00434077516919755157598282934, 2.21905934986599851932169852405, 2.46657337848958541233786904618, 2.59148655381906762217474663683, 3.03532685951216845715275839283, 3.22121415116759283032440267067, 3.28763863480721116545096077530, 3.50641892894935739851514565752, 3.84868474186927879830908423001, 4.44123371708524621083614394267, 4.46793328136069584504190941734, 4.68605182707309679630770282207, 5.03128012492627329283061143280, 5.10416049686817180215745340848, 5.21415148062023115787017345883, 5.40996499257468715029139143785, 5.72401450927163662784671482604, 5.83807579188317149847073664613, 6.07621847382744094365184216419, 6.19462237036591940926645435560

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

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