Properties

Degree 14
Conductor $ 1 $
Sign $-1$
Motivic weight 93
Primitive no
Self-dual yes
Analytic rank 7

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 4.37e13·2-s − 3.67e21·3-s − 1.49e28·4-s − 2.42e32·5-s − 1.60e35·6-s − 9.21e38·7-s − 6.18e41·8-s − 6.36e44·9-s − 1.06e46·10-s + 1.08e48·11-s + 5.51e49·12-s + 1.92e51·13-s − 4.02e52·14-s + 8.92e53·15-s + 2.21e55·16-s + 8.09e56·17-s − 2.78e58·18-s − 4.91e59·19-s + 3.63e60·20-s + 3.38e60·21-s + 4.73e61·22-s − 2.59e63·23-s + 2.27e63·24-s − 2.31e65·25-s + 8.41e64·26-s − 8.54e65·27-s + 1.38e67·28-s + ⋯
L(s)  = 1  + 0.439·2-s − 0.239·3-s − 1.51·4-s − 0.763·5-s − 0.105·6-s − 0.464·7-s − 0.627·8-s − 2.70·9-s − 0.335·10-s + 0.406·11-s + 0.362·12-s + 0.306·13-s − 0.204·14-s + 0.182·15-s + 0.225·16-s + 0.492·17-s − 1.18·18-s − 1.69·19-s + 1.15·20-s + 0.111·21-s + 0.178·22-s − 1.24·23-s + 0.150·24-s − 2.29·25-s + 0.134·26-s − 0.236·27-s + 0.703·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(94-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+93/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(14\)
\( N \)  =  \(1\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(93\)
character  :  $\chi_{1} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(7\)
Selberg data  =  \((14,\ 1,\ (\ :[93/2]^{7}),\ -1)\)
\(L(47)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{95}{2})\)   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 \( 1 - 683366042403 p^{6} T + \)\(41\!\cdots\!17\)\( p^{12} T^{2} - \)\(57\!\cdots\!35\)\( p^{27} T^{3} + \)\(21\!\cdots\!47\)\( p^{40} T^{4} - \)\(89\!\cdots\!83\)\( p^{63} T^{5} + \)\(39\!\cdots\!31\)\( p^{86} T^{6} - \)\(26\!\cdots\!05\)\( p^{115} T^{7} + \)\(39\!\cdots\!31\)\( p^{179} T^{8} - \)\(89\!\cdots\!83\)\( p^{249} T^{9} + \)\(21\!\cdots\!47\)\( p^{319} T^{10} - \)\(57\!\cdots\!35\)\( p^{399} T^{11} + \)\(41\!\cdots\!17\)\( p^{477} T^{12} - 683366042403 p^{564} T^{13} + p^{651} T^{14} \)
3 \( 1 + \)\(13\!\cdots\!92\)\( p^{3} T + \)\(33\!\cdots\!91\)\( p^{9} T^{2} + \)\(14\!\cdots\!80\)\( p^{18} T^{3} + \)\(12\!\cdots\!73\)\( p^{30} T^{4} + \)\(92\!\cdots\!56\)\( p^{45} T^{5} + \)\(21\!\cdots\!61\)\( p^{64} T^{6} + \)\(21\!\cdots\!80\)\( p^{83} T^{7} + \)\(21\!\cdots\!61\)\( p^{157} T^{8} + \)\(92\!\cdots\!56\)\( p^{231} T^{9} + \)\(12\!\cdots\!73\)\( p^{309} T^{10} + \)\(14\!\cdots\!80\)\( p^{390} T^{11} + \)\(33\!\cdots\!91\)\( p^{474} T^{12} + \)\(13\!\cdots\!92\)\( p^{561} T^{13} + p^{651} T^{14} \)
5 \( 1 + \)\(19\!\cdots\!38\)\( p^{3} T + \)\(74\!\cdots\!23\)\( p^{8} T^{2} + \)\(16\!\cdots\!44\)\( p^{14} T^{3} + \)\(13\!\cdots\!37\)\( p^{21} T^{4} + \)\(20\!\cdots\!98\)\( p^{30} T^{5} + \)\(75\!\cdots\!23\)\( p^{43} T^{6} + \)\(71\!\cdots\!52\)\( p^{58} T^{7} + \)\(75\!\cdots\!23\)\( p^{136} T^{8} + \)\(20\!\cdots\!98\)\( p^{216} T^{9} + \)\(13\!\cdots\!37\)\( p^{300} T^{10} + \)\(16\!\cdots\!44\)\( p^{386} T^{11} + \)\(74\!\cdots\!23\)\( p^{473} T^{12} + \)\(19\!\cdots\!38\)\( p^{561} T^{13} + p^{651} T^{14} \)
7 \( 1 + \)\(18\!\cdots\!92\)\( p^{2} T + \)\(10\!\cdots\!93\)\( p^{6} T^{2} - \)\(11\!\cdots\!00\)\( p^{10} T^{3} + \)\(11\!\cdots\!53\)\( p^{14} T^{4} - \)\(84\!\cdots\!24\)\( p^{20} T^{5} + \)\(12\!\cdots\!47\)\( p^{29} T^{6} - \)\(33\!\cdots\!00\)\( p^{39} T^{7} + \)\(12\!\cdots\!47\)\( p^{122} T^{8} - \)\(84\!\cdots\!24\)\( p^{206} T^{9} + \)\(11\!\cdots\!53\)\( p^{293} T^{10} - \)\(11\!\cdots\!00\)\( p^{382} T^{11} + \)\(10\!\cdots\!93\)\( p^{471} T^{12} + \)\(18\!\cdots\!92\)\( p^{560} T^{13} + p^{651} T^{14} \)
11 \( 1 - \)\(98\!\cdots\!84\)\( p T + \)\(26\!\cdots\!01\)\( p^{2} T^{2} - \)\(25\!\cdots\!44\)\( p^{5} T^{3} + \)\(20\!\cdots\!71\)\( p^{9} T^{4} - \)\(17\!\cdots\!72\)\( p^{14} T^{5} + \)\(70\!\cdots\!73\)\( p^{20} T^{6} - \)\(49\!\cdots\!72\)\( p^{26} T^{7} + \)\(70\!\cdots\!73\)\( p^{113} T^{8} - \)\(17\!\cdots\!72\)\( p^{200} T^{9} + \)\(20\!\cdots\!71\)\( p^{288} T^{10} - \)\(25\!\cdots\!44\)\( p^{377} T^{11} + \)\(26\!\cdots\!01\)\( p^{467} T^{12} - \)\(98\!\cdots\!84\)\( p^{559} T^{13} + p^{651} T^{14} \)
13 \( 1 - \)\(14\!\cdots\!02\)\( p T + \)\(78\!\cdots\!79\)\( p^{3} T^{2} - \)\(78\!\cdots\!40\)\( p^{6} T^{3} + \)\(10\!\cdots\!93\)\( p^{10} T^{4} - \)\(84\!\cdots\!18\)\( p^{14} T^{5} + \)\(76\!\cdots\!59\)\( p^{18} T^{6} - \)\(40\!\cdots\!40\)\( p^{23} T^{7} + \)\(76\!\cdots\!59\)\( p^{111} T^{8} - \)\(84\!\cdots\!18\)\( p^{200} T^{9} + \)\(10\!\cdots\!93\)\( p^{289} T^{10} - \)\(78\!\cdots\!40\)\( p^{378} T^{11} + \)\(78\!\cdots\!79\)\( p^{468} T^{12} - \)\(14\!\cdots\!02\)\( p^{559} T^{13} + p^{651} T^{14} \)
17 \( 1 - \)\(47\!\cdots\!26\)\( p T + \)\(39\!\cdots\!63\)\( p^{2} T^{2} - \)\(13\!\cdots\!40\)\( p^{4} T^{3} + \)\(15\!\cdots\!89\)\( p^{7} T^{4} - \)\(21\!\cdots\!18\)\( p^{11} T^{5} + \)\(80\!\cdots\!83\)\( p^{15} T^{6} - \)\(65\!\cdots\!20\)\( p^{20} T^{7} + \)\(80\!\cdots\!83\)\( p^{108} T^{8} - \)\(21\!\cdots\!18\)\( p^{197} T^{9} + \)\(15\!\cdots\!89\)\( p^{286} T^{10} - \)\(13\!\cdots\!40\)\( p^{376} T^{11} + \)\(39\!\cdots\!63\)\( p^{467} T^{12} - \)\(47\!\cdots\!26\)\( p^{559} T^{13} + p^{651} T^{14} \)
19 \( 1 + \)\(25\!\cdots\!00\)\( p T + \)\(11\!\cdots\!33\)\( p^{2} T^{2} + \)\(18\!\cdots\!00\)\( p^{3} T^{3} + \)\(41\!\cdots\!81\)\( p^{4} T^{4} + \)\(10\!\cdots\!00\)\( p^{7} T^{5} + \)\(55\!\cdots\!65\)\( p^{10} T^{6} + \)\(59\!\cdots\!00\)\( p^{14} T^{7} + \)\(55\!\cdots\!65\)\( p^{103} T^{8} + \)\(10\!\cdots\!00\)\( p^{193} T^{9} + \)\(41\!\cdots\!81\)\( p^{283} T^{10} + \)\(18\!\cdots\!00\)\( p^{375} T^{11} + \)\(11\!\cdots\!33\)\( p^{467} T^{12} + \)\(25\!\cdots\!00\)\( p^{559} T^{13} + p^{651} T^{14} \)
23 \( 1 + \)\(11\!\cdots\!68\)\( p T + \)\(29\!\cdots\!37\)\( p^{2} T^{2} + \)\(11\!\cdots\!60\)\( p^{4} T^{3} + \)\(40\!\cdots\!71\)\( p^{7} T^{4} + \)\(34\!\cdots\!48\)\( p^{8} T^{5} + \)\(20\!\cdots\!69\)\( p^{10} T^{6} + \)\(63\!\cdots\!80\)\( p^{12} T^{7} + \)\(20\!\cdots\!69\)\( p^{103} T^{8} + \)\(34\!\cdots\!48\)\( p^{194} T^{9} + \)\(40\!\cdots\!71\)\( p^{286} T^{10} + \)\(11\!\cdots\!60\)\( p^{376} T^{11} + \)\(29\!\cdots\!37\)\( p^{467} T^{12} + \)\(11\!\cdots\!68\)\( p^{559} T^{13} + p^{651} T^{14} \)
29 \( 1 - \)\(39\!\cdots\!50\)\( p T + \)\(24\!\cdots\!03\)\( p^{2} T^{2} - \)\(44\!\cdots\!00\)\( p^{4} T^{3} + \)\(74\!\cdots\!21\)\( p^{6} T^{4} - \)\(10\!\cdots\!50\)\( p^{8} T^{5} + \)\(47\!\cdots\!35\)\( p^{11} T^{6} - \)\(20\!\cdots\!00\)\( p^{14} T^{7} + \)\(47\!\cdots\!35\)\( p^{104} T^{8} - \)\(10\!\cdots\!50\)\( p^{194} T^{9} + \)\(74\!\cdots\!21\)\( p^{285} T^{10} - \)\(44\!\cdots\!00\)\( p^{376} T^{11} + \)\(24\!\cdots\!03\)\( p^{467} T^{12} - \)\(39\!\cdots\!50\)\( p^{559} T^{13} + p^{651} T^{14} \)
31 \( 1 + \)\(37\!\cdots\!76\)\( p T + \)\(12\!\cdots\!21\)\( p^{2} T^{2} + \)\(11\!\cdots\!56\)\( p^{3} T^{3} + \)\(84\!\cdots\!61\)\( p^{4} T^{4} + \)\(78\!\cdots\!48\)\( p^{5} T^{5} + \)\(73\!\cdots\!73\)\( p^{6} T^{6} + \)\(32\!\cdots\!08\)\( p^{7} T^{7} + \)\(73\!\cdots\!73\)\( p^{99} T^{8} + \)\(78\!\cdots\!48\)\( p^{191} T^{9} + \)\(84\!\cdots\!61\)\( p^{283} T^{10} + \)\(11\!\cdots\!56\)\( p^{375} T^{11} + \)\(12\!\cdots\!21\)\( p^{467} T^{12} + \)\(37\!\cdots\!76\)\( p^{559} T^{13} + p^{651} T^{14} \)
37 \( 1 - \)\(11\!\cdots\!42\)\( T + \)\(11\!\cdots\!11\)\( p T^{2} - \)\(27\!\cdots\!80\)\( p^{2} T^{3} + \)\(15\!\cdots\!49\)\( p^{3} T^{4} - \)\(80\!\cdots\!62\)\( p^{5} T^{5} + \)\(91\!\cdots\!03\)\( p^{7} T^{6} - \)\(38\!\cdots\!80\)\( p^{9} T^{7} + \)\(91\!\cdots\!03\)\( p^{100} T^{8} - \)\(80\!\cdots\!62\)\( p^{191} T^{9} + \)\(15\!\cdots\!49\)\( p^{282} T^{10} - \)\(27\!\cdots\!80\)\( p^{374} T^{11} + \)\(11\!\cdots\!11\)\( p^{466} T^{12} - \)\(11\!\cdots\!42\)\( p^{558} T^{13} + p^{651} T^{14} \)
41 \( 1 + \)\(12\!\cdots\!06\)\( p T + \)\(17\!\cdots\!31\)\( p^{2} T^{2} + \)\(44\!\cdots\!96\)\( p^{3} T^{3} + \)\(18\!\cdots\!81\)\( p^{4} T^{4} + \)\(11\!\cdots\!78\)\( p^{6} T^{5} + \)\(99\!\cdots\!23\)\( p^{8} T^{6} + \)\(42\!\cdots\!28\)\( p^{10} T^{7} + \)\(99\!\cdots\!23\)\( p^{101} T^{8} + \)\(11\!\cdots\!78\)\( p^{192} T^{9} + \)\(18\!\cdots\!81\)\( p^{283} T^{10} + \)\(44\!\cdots\!96\)\( p^{375} T^{11} + \)\(17\!\cdots\!31\)\( p^{467} T^{12} + \)\(12\!\cdots\!06\)\( p^{559} T^{13} + p^{651} T^{14} \)
43 \( 1 + \)\(16\!\cdots\!08\)\( p T + \)\(21\!\cdots\!57\)\( p^{2} T^{2} + \)\(37\!\cdots\!00\)\( p^{3} T^{3} + \)\(21\!\cdots\!97\)\( p^{4} T^{4} + \)\(87\!\cdots\!32\)\( p^{6} T^{5} + \)\(74\!\cdots\!21\)\( p^{8} T^{6} + \)\(26\!\cdots\!00\)\( p^{10} T^{7} + \)\(74\!\cdots\!21\)\( p^{101} T^{8} + \)\(87\!\cdots\!32\)\( p^{192} T^{9} + \)\(21\!\cdots\!97\)\( p^{283} T^{10} + \)\(37\!\cdots\!00\)\( p^{375} T^{11} + \)\(21\!\cdots\!57\)\( p^{467} T^{12} + \)\(16\!\cdots\!08\)\( p^{559} T^{13} + p^{651} T^{14} \)
47 \( 1 + \)\(37\!\cdots\!08\)\( T + \)\(11\!\cdots\!57\)\( T^{2} + \)\(50\!\cdots\!40\)\( T^{3} + \)\(66\!\cdots\!97\)\( T^{4} + \)\(64\!\cdots\!68\)\( p T^{5} + \)\(13\!\cdots\!21\)\( p^{2} T^{6} + \)\(11\!\cdots\!40\)\( p^{3} T^{7} + \)\(13\!\cdots\!21\)\( p^{95} T^{8} + \)\(64\!\cdots\!68\)\( p^{187} T^{9} + \)\(66\!\cdots\!97\)\( p^{279} T^{10} + \)\(50\!\cdots\!40\)\( p^{372} T^{11} + \)\(11\!\cdots\!57\)\( p^{465} T^{12} + \)\(37\!\cdots\!08\)\( p^{558} T^{13} + p^{651} T^{14} \)
53 \( 1 + \)\(36\!\cdots\!34\)\( T + \)\(10\!\cdots\!03\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(74\!\cdots\!09\)\( p T^{4} + \)\(23\!\cdots\!62\)\( p^{2} T^{5} + \)\(88\!\cdots\!83\)\( p^{3} T^{6} + \)\(25\!\cdots\!60\)\( p^{4} T^{7} + \)\(88\!\cdots\!83\)\( p^{96} T^{8} + \)\(23\!\cdots\!62\)\( p^{188} T^{9} + \)\(74\!\cdots\!09\)\( p^{280} T^{10} + \)\(19\!\cdots\!20\)\( p^{372} T^{11} + \)\(10\!\cdots\!03\)\( p^{465} T^{12} + \)\(36\!\cdots\!34\)\( p^{558} T^{13} + p^{651} T^{14} \)
59 \( 1 + \)\(11\!\cdots\!00\)\( T + \)\(87\!\cdots\!53\)\( T^{2} + \)\(45\!\cdots\!00\)\( T^{3} + \)\(32\!\cdots\!79\)\( p T^{4} + \)\(18\!\cdots\!00\)\( p^{2} T^{5} + \)\(88\!\cdots\!35\)\( p^{3} T^{6} + \)\(35\!\cdots\!00\)\( p^{4} T^{7} + \)\(88\!\cdots\!35\)\( p^{96} T^{8} + \)\(18\!\cdots\!00\)\( p^{188} T^{9} + \)\(32\!\cdots\!79\)\( p^{280} T^{10} + \)\(45\!\cdots\!00\)\( p^{372} T^{11} + \)\(87\!\cdots\!53\)\( p^{465} T^{12} + \)\(11\!\cdots\!00\)\( p^{558} T^{13} + p^{651} T^{14} \)
61 \( 1 + \)\(32\!\cdots\!26\)\( T + \)\(80\!\cdots\!71\)\( T^{2} + \)\(23\!\cdots\!96\)\( p T^{3} + \)\(57\!\cdots\!41\)\( p^{2} T^{4} + \)\(11\!\cdots\!58\)\( p^{3} T^{5} + \)\(22\!\cdots\!03\)\( p^{4} T^{6} + \)\(39\!\cdots\!08\)\( p^{5} T^{7} + \)\(22\!\cdots\!03\)\( p^{97} T^{8} + \)\(11\!\cdots\!58\)\( p^{189} T^{9} + \)\(57\!\cdots\!41\)\( p^{281} T^{10} + \)\(23\!\cdots\!96\)\( p^{373} T^{11} + \)\(80\!\cdots\!71\)\( p^{465} T^{12} + \)\(32\!\cdots\!26\)\( p^{558} T^{13} + p^{651} T^{14} \)
67 \( 1 - \)\(97\!\cdots\!92\)\( T + \)\(34\!\cdots\!57\)\( T^{2} - \)\(42\!\cdots\!20\)\( p T^{3} + \)\(12\!\cdots\!73\)\( p^{2} T^{4} - \)\(13\!\cdots\!88\)\( p^{3} T^{5} + \)\(26\!\cdots\!89\)\( p^{4} T^{6} - \)\(24\!\cdots\!60\)\( p^{5} T^{7} + \)\(26\!\cdots\!89\)\( p^{97} T^{8} - \)\(13\!\cdots\!88\)\( p^{189} T^{9} + \)\(12\!\cdots\!73\)\( p^{281} T^{10} - \)\(42\!\cdots\!20\)\( p^{373} T^{11} + \)\(34\!\cdots\!57\)\( p^{465} T^{12} - \)\(97\!\cdots\!92\)\( p^{558} T^{13} + p^{651} T^{14} \)
71 \( 1 - \)\(42\!\cdots\!84\)\( T + \)\(12\!\cdots\!01\)\( T^{2} - \)\(37\!\cdots\!44\)\( p T^{3} + \)\(92\!\cdots\!81\)\( p^{2} T^{4} - \)\(18\!\cdots\!32\)\( p^{3} T^{5} + \)\(35\!\cdots\!53\)\( p^{4} T^{6} - \)\(60\!\cdots\!52\)\( p^{5} T^{7} + \)\(35\!\cdots\!53\)\( p^{97} T^{8} - \)\(18\!\cdots\!32\)\( p^{189} T^{9} + \)\(92\!\cdots\!81\)\( p^{281} T^{10} - \)\(37\!\cdots\!44\)\( p^{373} T^{11} + \)\(12\!\cdots\!01\)\( p^{465} T^{12} - \)\(42\!\cdots\!84\)\( p^{558} T^{13} + p^{651} T^{14} \)
73 \( 1 - \)\(24\!\cdots\!86\)\( T + \)\(97\!\cdots\!51\)\( p T^{2} - \)\(43\!\cdots\!60\)\( p^{2} T^{3} + \)\(75\!\cdots\!61\)\( p^{3} T^{4} - \)\(44\!\cdots\!34\)\( p^{5} T^{5} + \)\(39\!\cdots\!67\)\( p^{5} T^{6} - \)\(14\!\cdots\!80\)\( p^{6} T^{7} + \)\(39\!\cdots\!67\)\( p^{98} T^{8} - \)\(44\!\cdots\!34\)\( p^{191} T^{9} + \)\(75\!\cdots\!61\)\( p^{282} T^{10} - \)\(43\!\cdots\!60\)\( p^{374} T^{11} + \)\(97\!\cdots\!51\)\( p^{466} T^{12} - \)\(24\!\cdots\!86\)\( p^{558} T^{13} + p^{651} T^{14} \)
79 \( 1 + \)\(43\!\cdots\!00\)\( T + \)\(19\!\cdots\!87\)\( p T^{2} + \)\(58\!\cdots\!00\)\( p^{2} T^{3} + \)\(14\!\cdots\!19\)\( p^{3} T^{4} + \)\(28\!\cdots\!00\)\( p^{4} T^{5} + \)\(52\!\cdots\!35\)\( p^{5} T^{6} + \)\(10\!\cdots\!00\)\( p^{6} T^{7} + \)\(52\!\cdots\!35\)\( p^{98} T^{8} + \)\(28\!\cdots\!00\)\( p^{190} T^{9} + \)\(14\!\cdots\!19\)\( p^{282} T^{10} + \)\(58\!\cdots\!00\)\( p^{374} T^{11} + \)\(19\!\cdots\!87\)\( p^{466} T^{12} + \)\(43\!\cdots\!00\)\( p^{558} T^{13} + p^{651} T^{14} \)
83 \( 1 + \)\(20\!\cdots\!04\)\( T + \)\(18\!\cdots\!51\)\( p T^{2} + \)\(43\!\cdots\!20\)\( p^{2} T^{3} + \)\(20\!\cdots\!91\)\( p^{3} T^{4} + \)\(40\!\cdots\!68\)\( p^{4} T^{5} + \)\(13\!\cdots\!07\)\( p^{5} T^{6} + \)\(21\!\cdots\!60\)\( p^{6} T^{7} + \)\(13\!\cdots\!07\)\( p^{98} T^{8} + \)\(40\!\cdots\!68\)\( p^{190} T^{9} + \)\(20\!\cdots\!91\)\( p^{282} T^{10} + \)\(43\!\cdots\!20\)\( p^{374} T^{11} + \)\(18\!\cdots\!51\)\( p^{466} T^{12} + \)\(20\!\cdots\!04\)\( p^{558} T^{13} + p^{651} T^{14} \)
89 \( 1 - \)\(62\!\cdots\!50\)\( p T + \)\(16\!\cdots\!23\)\( p^{2} T^{2} - \)\(88\!\cdots\!00\)\( p^{3} T^{3} + \)\(11\!\cdots\!41\)\( p^{4} T^{4} - \)\(53\!\cdots\!50\)\( p^{5} T^{5} + \)\(47\!\cdots\!15\)\( p^{6} T^{6} - \)\(17\!\cdots\!00\)\( p^{7} T^{7} + \)\(47\!\cdots\!15\)\( p^{99} T^{8} - \)\(53\!\cdots\!50\)\( p^{191} T^{9} + \)\(11\!\cdots\!41\)\( p^{283} T^{10} - \)\(88\!\cdots\!00\)\( p^{375} T^{11} + \)\(16\!\cdots\!23\)\( p^{467} T^{12} - \)\(62\!\cdots\!50\)\( p^{559} T^{13} + p^{651} T^{14} \)
97 \( 1 - \)\(43\!\cdots\!42\)\( T + \)\(20\!\cdots\!07\)\( T^{2} - \)\(46\!\cdots\!60\)\( T^{3} + \)\(22\!\cdots\!97\)\( T^{4} + \)\(91\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!39\)\( T^{6} + \)\(19\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!39\)\( p^{93} T^{8} + \)\(91\!\cdots\!46\)\( p^{186} T^{9} + \)\(22\!\cdots\!97\)\( p^{279} T^{10} - \)\(46\!\cdots\!60\)\( p^{372} T^{11} + \)\(20\!\cdots\!07\)\( p^{465} T^{12} - \)\(43\!\cdots\!42\)\( p^{558} T^{13} + p^{651} T^{14} \)
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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

−6.12719171466360395466622317663, −6.08732055577577117011239224737, −6.08620355525781336302059505149, −5.61204218607483688429891194675, −5.15604176592037110145585460923, −5.05429458047663572858791949171, −5.00516304517308193266802192254, −4.68893646141232147844535620846, −4.44286336617295541313490891036, −4.32304324376174698821086174483, −4.09951250907526273424642340885, −3.63339109649557768810454088784, −3.60911542290329273613954173090, −3.57669082731974987690155176724, −3.19749320409605204402321686184, −3.07628073342446657369249799081, −2.69238081254357342412253396197, −2.56495704095559300976348074634, −2.37613694941362268153927827413, −1.93543058059972283252551390914, −1.76598384822919589294929158602, −1.64982979947534669826343267758, −1.24799338765042645218901883671, −1.12993969144782936120472009162, −0.957896353390260094617286693473, 0, 0, 0, 0, 0, 0, 0, 0.957896353390260094617286693473, 1.12993969144782936120472009162, 1.24799338765042645218901883671, 1.64982979947534669826343267758, 1.76598384822919589294929158602, 1.93543058059972283252551390914, 2.37613694941362268153927827413, 2.56495704095559300976348074634, 2.69238081254357342412253396197, 3.07628073342446657369249799081, 3.19749320409605204402321686184, 3.57669082731974987690155176724, 3.60911542290329273613954173090, 3.63339109649557768810454088784, 4.09951250907526273424642340885, 4.32304324376174698821086174483, 4.44286336617295541313490891036, 4.68893646141232147844535620846, 5.00516304517308193266802192254, 5.05429458047663572858791949171, 5.15604176592037110145585460923, 5.61204218607483688429891194675, 6.08620355525781336302059505149, 6.08732055577577117011239224737, 6.12719171466360395466622317663

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

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