Properties

Label 20-91e10-1.1-c7e10-0-1
Degree $20$
Conductor $3.894\times 10^{19}$
Sign $1$
Analytic cond. $3.44600\times 10^{14}$
Root an. cond. $5.33170$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $10$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 18·2-s − 80·3-s − 143·4-s − 927·5-s + 1.44e3·6-s + 3.43e3·7-s + 3.66e3·8-s − 5.92e3·9-s + 1.66e4·10-s + 876·11-s + 1.14e4·12-s − 2.19e4·13-s − 6.17e4·14-s + 7.41e4·15-s + 5.82e3·16-s + 6.29e3·17-s + 1.06e5·18-s − 9.74e4·19-s + 1.32e5·20-s − 2.74e5·21-s − 1.57e4·22-s − 1.52e4·23-s − 2.92e5·24-s + 1.20e5·25-s + 3.95e5·26-s + 6.42e5·27-s − 4.90e5·28-s + ⋯
L(s)  = 1  − 1.59·2-s − 1.71·3-s − 1.11·4-s − 3.31·5-s + 2.72·6-s + 3.77·7-s + 2.52·8-s − 2.71·9-s + 5.27·10-s + 0.198·11-s + 1.91·12-s − 2.77·13-s − 6.01·14-s + 5.67·15-s + 0.355·16-s + 0.310·17-s + 4.31·18-s − 3.25·19-s + 3.70·20-s − 6.46·21-s − 0.315·22-s − 0.261·23-s − 4.32·24-s + 1.53·25-s + 4.41·26-s + 6.28·27-s − 4.22·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{10} \cdot 13^{10}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{10} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(7^{10} \cdot 13^{10}\)
Sign: $1$
Analytic conductor: \(3.44600\times 10^{14}\)
Root analytic conductor: \(5.33170\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{91} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(10\)
Selberg data: \((20,\ 7^{10} \cdot 13^{10} ,\ ( \ : [7/2]^{10} ),\ 1 )\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( ( 1 - p^{3} T )^{10} \)
13 \( ( 1 + p^{3} T )^{10} \)
good2 \( 1 + 9 p T + 467 T^{2} + 915 p^{3} T^{3} + 63419 p T^{4} + 438387 p^{2} T^{5} + 760589 p^{5} T^{6} + 9460809 p^{5} T^{7} + 115938499 p^{5} T^{8} + 677515623 p^{6} T^{9} + 987531001 p^{9} T^{10} + 677515623 p^{13} T^{11} + 115938499 p^{19} T^{12} + 9460809 p^{26} T^{13} + 760589 p^{33} T^{14} + 438387 p^{37} T^{15} + 63419 p^{43} T^{16} + 915 p^{52} T^{17} + 467 p^{56} T^{18} + 9 p^{64} T^{19} + p^{70} T^{20} \)
3 \( 1 + 80 T + 12329 T^{2} + 817940 T^{3} + 71843209 T^{4} + 1312521356 p T^{5} + 3242643040 p^{4} T^{6} + 455056226236 p^{3} T^{7} + 108374921278 p^{8} T^{8} + 123141224387804 p^{5} T^{9} + 2244009932158174 p^{6} T^{10} + 123141224387804 p^{12} T^{11} + 108374921278 p^{22} T^{12} + 455056226236 p^{24} T^{13} + 3242643040 p^{32} T^{14} + 1312521356 p^{36} T^{15} + 71843209 p^{42} T^{16} + 817940 p^{49} T^{17} + 12329 p^{56} T^{18} + 80 p^{63} T^{19} + p^{70} T^{20} \)
5 \( 1 + 927 T + 739217 T^{2} + 391941558 T^{3} + 7911620286 p^{2} T^{4} + 3269820145446 p^{2} T^{5} + 263991052184879 p^{3} T^{6} + 18454924806030873 p^{4} T^{7} + 251846002159619219 p^{6} T^{8} + 76025942809838711604 p^{6} T^{9} + \)\(44\!\cdots\!76\)\( p^{7} T^{10} + 76025942809838711604 p^{13} T^{11} + 251846002159619219 p^{20} T^{12} + 18454924806030873 p^{25} T^{13} + 263991052184879 p^{31} T^{14} + 3269820145446 p^{37} T^{15} + 7911620286 p^{44} T^{16} + 391941558 p^{49} T^{17} + 739217 p^{56} T^{18} + 927 p^{63} T^{19} + p^{70} T^{20} \)
11 \( 1 - 876 T + 84892563 T^{2} - 55846947570 T^{3} + 3511199977450925 T^{4} - 2711527102283681982 T^{5} + \)\(92\!\cdots\!52\)\( T^{6} - \)\(97\!\cdots\!50\)\( T^{7} + \)\(16\!\cdots\!94\)\( p T^{8} - \)\(26\!\cdots\!22\)\( T^{9} + \)\(33\!\cdots\!14\)\( T^{10} - \)\(26\!\cdots\!22\)\( p^{7} T^{11} + \)\(16\!\cdots\!94\)\( p^{15} T^{12} - \)\(97\!\cdots\!50\)\( p^{21} T^{13} + \)\(92\!\cdots\!52\)\( p^{28} T^{14} - 2711527102283681982 p^{35} T^{15} + 3511199977450925 p^{42} T^{16} - 55846947570 p^{49} T^{17} + 84892563 p^{56} T^{18} - 876 p^{63} T^{19} + p^{70} T^{20} \)
17 \( 1 - 6294 T + 1130710490 T^{2} - 8696007265734 T^{3} + 1020298066655460913 T^{4} - \)\(61\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!96\)\( T^{6} - \)\(37\!\cdots\!44\)\( T^{7} + \)\(35\!\cdots\!98\)\( T^{8} - \)\(18\!\cdots\!64\)\( T^{9} + \)\(15\!\cdots\!32\)\( T^{10} - \)\(18\!\cdots\!64\)\( p^{7} T^{11} + \)\(35\!\cdots\!98\)\( p^{14} T^{12} - \)\(37\!\cdots\!44\)\( p^{21} T^{13} + \)\(64\!\cdots\!96\)\( p^{28} T^{14} - \)\(61\!\cdots\!56\)\( p^{35} T^{15} + 1020298066655460913 p^{42} T^{16} - 8696007265734 p^{49} T^{17} + 1130710490 p^{56} T^{18} - 6294 p^{63} T^{19} + p^{70} T^{20} \)
19 \( 1 + 97401 T + 10058860839 T^{2} + 668514864618420 T^{3} + 42044509772596922984 T^{4} + \)\(21\!\cdots\!64\)\( T^{5} + \)\(10\!\cdots\!29\)\( T^{6} + \)\(41\!\cdots\!21\)\( T^{7} + \)\(15\!\cdots\!39\)\( T^{8} + \)\(53\!\cdots\!88\)\( T^{9} + \)\(16\!\cdots\!48\)\( T^{10} + \)\(53\!\cdots\!88\)\( p^{7} T^{11} + \)\(15\!\cdots\!39\)\( p^{14} T^{12} + \)\(41\!\cdots\!21\)\( p^{21} T^{13} + \)\(10\!\cdots\!29\)\( p^{28} T^{14} + \)\(21\!\cdots\!64\)\( p^{35} T^{15} + 42044509772596922984 p^{42} T^{16} + 668514864618420 p^{49} T^{17} + 10058860839 p^{56} T^{18} + 97401 p^{63} T^{19} + p^{70} T^{20} \)
23 \( 1 + 15255 T + 19696234850 T^{2} + 148057171659861 T^{3} + \)\(18\!\cdots\!47\)\( T^{4} + \)\(16\!\cdots\!86\)\( T^{5} + \)\(10\!\cdots\!88\)\( T^{6} - \)\(44\!\cdots\!10\)\( T^{7} + \)\(47\!\cdots\!61\)\( T^{8} - \)\(32\!\cdots\!15\)\( T^{9} + \)\(17\!\cdots\!14\)\( T^{10} - \)\(32\!\cdots\!15\)\( p^{7} T^{11} + \)\(47\!\cdots\!61\)\( p^{14} T^{12} - \)\(44\!\cdots\!10\)\( p^{21} T^{13} + \)\(10\!\cdots\!88\)\( p^{28} T^{14} + \)\(16\!\cdots\!86\)\( p^{35} T^{15} + \)\(18\!\cdots\!47\)\( p^{42} T^{16} + 148057171659861 p^{49} T^{17} + 19696234850 p^{56} T^{18} + 15255 p^{63} T^{19} + p^{70} T^{20} \)
29 \( 1 + 340533 T + 162292724693 T^{2} + 42295312323668946 T^{3} + \)\(11\!\cdots\!74\)\( T^{4} + \)\(24\!\cdots\!54\)\( T^{5} + \)\(49\!\cdots\!43\)\( T^{6} + \)\(87\!\cdots\!51\)\( T^{7} + \)\(14\!\cdots\!19\)\( T^{8} + \)\(21\!\cdots\!68\)\( T^{9} + \)\(29\!\cdots\!52\)\( T^{10} + \)\(21\!\cdots\!68\)\( p^{7} T^{11} + \)\(14\!\cdots\!19\)\( p^{14} T^{12} + \)\(87\!\cdots\!51\)\( p^{21} T^{13} + \)\(49\!\cdots\!43\)\( p^{28} T^{14} + \)\(24\!\cdots\!54\)\( p^{35} T^{15} + \)\(11\!\cdots\!74\)\( p^{42} T^{16} + 42295312323668946 p^{49} T^{17} + 162292724693 p^{56} T^{18} + 340533 p^{63} T^{19} + p^{70} T^{20} \)
31 \( 1 + 148675 T + 138661334430 T^{2} + 11636548791078637 T^{3} + \)\(86\!\cdots\!19\)\( T^{4} + \)\(18\!\cdots\!58\)\( T^{5} + \)\(32\!\cdots\!00\)\( T^{6} - \)\(50\!\cdots\!86\)\( p T^{7} + \)\(88\!\cdots\!69\)\( T^{8} - \)\(10\!\cdots\!67\)\( T^{9} + \)\(22\!\cdots\!70\)\( T^{10} - \)\(10\!\cdots\!67\)\( p^{7} T^{11} + \)\(88\!\cdots\!69\)\( p^{14} T^{12} - \)\(50\!\cdots\!86\)\( p^{22} T^{13} + \)\(32\!\cdots\!00\)\( p^{28} T^{14} + \)\(18\!\cdots\!58\)\( p^{35} T^{15} + \)\(86\!\cdots\!19\)\( p^{42} T^{16} + 11636548791078637 p^{49} T^{17} + 138661334430 p^{56} T^{18} + 148675 p^{63} T^{19} + p^{70} T^{20} \)
37 \( 1 + 621782 T + 698014408413 T^{2} + 7742878903630964 p T^{3} + \)\(18\!\cdots\!37\)\( T^{4} + \)\(52\!\cdots\!78\)\( T^{5} + \)\(25\!\cdots\!16\)\( T^{6} + \)\(47\!\cdots\!38\)\( T^{7} + \)\(62\!\cdots\!74\)\( p T^{8} + \)\(24\!\cdots\!78\)\( T^{9} + \)\(19\!\cdots\!66\)\( T^{10} + \)\(24\!\cdots\!78\)\( p^{7} T^{11} + \)\(62\!\cdots\!74\)\( p^{15} T^{12} + \)\(47\!\cdots\!38\)\( p^{21} T^{13} + \)\(25\!\cdots\!16\)\( p^{28} T^{14} + \)\(52\!\cdots\!78\)\( p^{35} T^{15} + \)\(18\!\cdots\!37\)\( p^{42} T^{16} + 7742878903630964 p^{50} T^{17} + 698014408413 p^{56} T^{18} + 621782 p^{63} T^{19} + p^{70} T^{20} \)
41 \( 1 + 2043336 T + 2427151728769 T^{2} + 2006971950576044208 T^{3} + \)\(13\!\cdots\!09\)\( T^{4} + \)\(73\!\cdots\!36\)\( T^{5} + \)\(37\!\cdots\!08\)\( T^{6} + \)\(18\!\cdots\!04\)\( T^{7} + \)\(92\!\cdots\!70\)\( T^{8} + \)\(45\!\cdots\!32\)\( T^{9} + \)\(21\!\cdots\!50\)\( T^{10} + \)\(45\!\cdots\!32\)\( p^{7} T^{11} + \)\(92\!\cdots\!70\)\( p^{14} T^{12} + \)\(18\!\cdots\!04\)\( p^{21} T^{13} + \)\(37\!\cdots\!08\)\( p^{28} T^{14} + \)\(73\!\cdots\!36\)\( p^{35} T^{15} + \)\(13\!\cdots\!09\)\( p^{42} T^{16} + 2006971950576044208 p^{49} T^{17} + 2427151728769 p^{56} T^{18} + 2043336 p^{63} T^{19} + p^{70} T^{20} \)
43 \( 1 + 1801391 T + 2132391745167 T^{2} + 2037123852742583696 T^{3} + \)\(17\!\cdots\!36\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(95\!\cdots\!45\)\( T^{6} + \)\(62\!\cdots\!51\)\( T^{7} + \)\(38\!\cdots\!19\)\( T^{8} + \)\(21\!\cdots\!24\)\( T^{9} + \)\(11\!\cdots\!88\)\( T^{10} + \)\(21\!\cdots\!24\)\( p^{7} T^{11} + \)\(38\!\cdots\!19\)\( p^{14} T^{12} + \)\(62\!\cdots\!51\)\( p^{21} T^{13} + \)\(95\!\cdots\!45\)\( p^{28} T^{14} + \)\(13\!\cdots\!32\)\( p^{35} T^{15} + \)\(17\!\cdots\!36\)\( p^{42} T^{16} + 2037123852742583696 p^{49} T^{17} + 2132391745167 p^{56} T^{18} + 1801391 p^{63} T^{19} + p^{70} T^{20} \)
47 \( 1 + 1624701 T + 4482864613330 T^{2} + 5661168346576340175 T^{3} + \)\(88\!\cdots\!11\)\( T^{4} + \)\(90\!\cdots\!46\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} + \)\(90\!\cdots\!94\)\( T^{7} + \)\(84\!\cdots\!05\)\( T^{8} + \)\(62\!\cdots\!79\)\( T^{9} + \)\(49\!\cdots\!58\)\( T^{10} + \)\(62\!\cdots\!79\)\( p^{7} T^{11} + \)\(84\!\cdots\!05\)\( p^{14} T^{12} + \)\(90\!\cdots\!94\)\( p^{21} T^{13} + \)\(10\!\cdots\!80\)\( p^{28} T^{14} + \)\(90\!\cdots\!46\)\( p^{35} T^{15} + \)\(88\!\cdots\!11\)\( p^{42} T^{16} + 5661168346576340175 p^{49} T^{17} + 4482864613330 p^{56} T^{18} + 1624701 p^{63} T^{19} + p^{70} T^{20} \)
53 \( 1 + 199965 T + 4737965863913 T^{2} - 592108304512241070 T^{3} + \)\(13\!\cdots\!54\)\( T^{4} - \)\(25\!\cdots\!70\)\( T^{5} + \)\(28\!\cdots\!15\)\( T^{6} - \)\(11\!\cdots\!17\)\( p T^{7} + \)\(47\!\cdots\!83\)\( T^{8} - \)\(10\!\cdots\!68\)\( T^{9} + \)\(62\!\cdots\!08\)\( T^{10} - \)\(10\!\cdots\!68\)\( p^{7} T^{11} + \)\(47\!\cdots\!83\)\( p^{14} T^{12} - \)\(11\!\cdots\!17\)\( p^{22} T^{13} + \)\(28\!\cdots\!15\)\( p^{28} T^{14} - \)\(25\!\cdots\!70\)\( p^{35} T^{15} + \)\(13\!\cdots\!54\)\( p^{42} T^{16} - 592108304512241070 p^{49} T^{17} + 4737965863913 p^{56} T^{18} + 199965 p^{63} T^{19} + p^{70} T^{20} \)
59 \( 1 + 8098908 T + 48823051368714 T^{2} + \)\(20\!\cdots\!32\)\( T^{3} + \)\(74\!\cdots\!37\)\( T^{4} + \)\(22\!\cdots\!96\)\( T^{5} + \)\(58\!\cdots\!12\)\( T^{6} + \)\(13\!\cdots\!40\)\( T^{7} + \)\(27\!\cdots\!06\)\( T^{8} + \)\(50\!\cdots\!84\)\( T^{9} + \)\(84\!\cdots\!56\)\( T^{10} + \)\(50\!\cdots\!84\)\( p^{7} T^{11} + \)\(27\!\cdots\!06\)\( p^{14} T^{12} + \)\(13\!\cdots\!40\)\( p^{21} T^{13} + \)\(58\!\cdots\!12\)\( p^{28} T^{14} + \)\(22\!\cdots\!96\)\( p^{35} T^{15} + \)\(74\!\cdots\!37\)\( p^{42} T^{16} + \)\(20\!\cdots\!32\)\( p^{49} T^{17} + 48823051368714 p^{56} T^{18} + 8098908 p^{63} T^{19} + p^{70} T^{20} \)
61 \( 1 - 2271618 T + 20415659278407 T^{2} - 44048246929000348806 T^{3} + \)\(21\!\cdots\!25\)\( T^{4} - \)\(42\!\cdots\!84\)\( T^{5} + \)\(14\!\cdots\!84\)\( T^{6} - \)\(25\!\cdots\!36\)\( T^{7} + \)\(72\!\cdots\!06\)\( T^{8} - \)\(11\!\cdots\!28\)\( T^{9} + \)\(26\!\cdots\!30\)\( T^{10} - \)\(11\!\cdots\!28\)\( p^{7} T^{11} + \)\(72\!\cdots\!06\)\( p^{14} T^{12} - \)\(25\!\cdots\!36\)\( p^{21} T^{13} + \)\(14\!\cdots\!84\)\( p^{28} T^{14} - \)\(42\!\cdots\!84\)\( p^{35} T^{15} + \)\(21\!\cdots\!25\)\( p^{42} T^{16} - 44048246929000348806 p^{49} T^{17} + 20415659278407 p^{56} T^{18} - 2271618 p^{63} T^{19} + p^{70} T^{20} \)
67 \( 1 - 1970272 T + 23359399960819 T^{2} - 32230927731301169530 T^{3} + \)\(24\!\cdots\!05\)\( T^{4} - \)\(17\!\cdots\!30\)\( T^{5} + \)\(22\!\cdots\!12\)\( p T^{6} + \)\(39\!\cdots\!66\)\( T^{7} + \)\(75\!\cdots\!06\)\( T^{8} + \)\(11\!\cdots\!10\)\( T^{9} + \)\(40\!\cdots\!46\)\( T^{10} + \)\(11\!\cdots\!10\)\( p^{7} T^{11} + \)\(75\!\cdots\!06\)\( p^{14} T^{12} + \)\(39\!\cdots\!66\)\( p^{21} T^{13} + \)\(22\!\cdots\!12\)\( p^{29} T^{14} - \)\(17\!\cdots\!30\)\( p^{35} T^{15} + \)\(24\!\cdots\!05\)\( p^{42} T^{16} - 32230927731301169530 p^{49} T^{17} + 23359399960819 p^{56} T^{18} - 1970272 p^{63} T^{19} + p^{70} T^{20} \)
71 \( 1 + 7145820 T + 68794567117590 T^{2} + \)\(32\!\cdots\!28\)\( T^{3} + \)\(19\!\cdots\!97\)\( T^{4} + \)\(72\!\cdots\!04\)\( T^{5} + \)\(34\!\cdots\!00\)\( T^{6} + \)\(11\!\cdots\!68\)\( T^{7} + \)\(45\!\cdots\!62\)\( T^{8} + \)\(12\!\cdots\!80\)\( T^{9} + \)\(45\!\cdots\!24\)\( T^{10} + \)\(12\!\cdots\!80\)\( p^{7} T^{11} + \)\(45\!\cdots\!62\)\( p^{14} T^{12} + \)\(11\!\cdots\!68\)\( p^{21} T^{13} + \)\(34\!\cdots\!00\)\( p^{28} T^{14} + \)\(72\!\cdots\!04\)\( p^{35} T^{15} + \)\(19\!\cdots\!97\)\( p^{42} T^{16} + \)\(32\!\cdots\!28\)\( p^{49} T^{17} + 68794567117590 p^{56} T^{18} + 7145820 p^{63} T^{19} + p^{70} T^{20} \)
73 \( 1 - 1409431 T + 55876749773232 T^{2} - 80476770795564131467 T^{3} + \)\(15\!\cdots\!31\)\( T^{4} - \)\(25\!\cdots\!02\)\( T^{5} + \)\(28\!\cdots\!36\)\( T^{6} - \)\(52\!\cdots\!38\)\( T^{7} + \)\(39\!\cdots\!21\)\( T^{8} - \)\(77\!\cdots\!49\)\( T^{9} + \)\(46\!\cdots\!36\)\( T^{10} - \)\(77\!\cdots\!49\)\( p^{7} T^{11} + \)\(39\!\cdots\!21\)\( p^{14} T^{12} - \)\(52\!\cdots\!38\)\( p^{21} T^{13} + \)\(28\!\cdots\!36\)\( p^{28} T^{14} - \)\(25\!\cdots\!02\)\( p^{35} T^{15} + \)\(15\!\cdots\!31\)\( p^{42} T^{16} - 80476770795564131467 p^{49} T^{17} + 55876749773232 p^{56} T^{18} - 1409431 p^{63} T^{19} + p^{70} T^{20} \)
79 \( 1 + 9011055 T + 154462368133602 T^{2} + \)\(11\!\cdots\!29\)\( T^{3} + \)\(11\!\cdots\!27\)\( T^{4} + \)\(68\!\cdots\!18\)\( T^{5} + \)\(49\!\cdots\!48\)\( T^{6} + \)\(26\!\cdots\!86\)\( T^{7} + \)\(15\!\cdots\!89\)\( T^{8} + \)\(70\!\cdots\!85\)\( T^{9} + \)\(33\!\cdots\!58\)\( T^{10} + \)\(70\!\cdots\!85\)\( p^{7} T^{11} + \)\(15\!\cdots\!89\)\( p^{14} T^{12} + \)\(26\!\cdots\!86\)\( p^{21} T^{13} + \)\(49\!\cdots\!48\)\( p^{28} T^{14} + \)\(68\!\cdots\!18\)\( p^{35} T^{15} + \)\(11\!\cdots\!27\)\( p^{42} T^{16} + \)\(11\!\cdots\!29\)\( p^{49} T^{17} + 154462368133602 p^{56} T^{18} + 9011055 p^{63} T^{19} + p^{70} T^{20} \)
83 \( 1 + 15006567 T + 259972667022383 T^{2} + \)\(27\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!04\)\( T^{4} + \)\(24\!\cdots\!12\)\( T^{5} + \)\(19\!\cdots\!49\)\( T^{6} + \)\(13\!\cdots\!27\)\( T^{7} + \)\(88\!\cdots\!31\)\( T^{8} + \)\(50\!\cdots\!60\)\( T^{9} + \)\(28\!\cdots\!40\)\( T^{10} + \)\(50\!\cdots\!60\)\( p^{7} T^{11} + \)\(88\!\cdots\!31\)\( p^{14} T^{12} + \)\(13\!\cdots\!27\)\( p^{21} T^{13} + \)\(19\!\cdots\!49\)\( p^{28} T^{14} + \)\(24\!\cdots\!12\)\( p^{35} T^{15} + \)\(29\!\cdots\!04\)\( p^{42} T^{16} + \)\(27\!\cdots\!80\)\( p^{49} T^{17} + 259972667022383 p^{56} T^{18} + 15006567 p^{63} T^{19} + p^{70} T^{20} \)
89 \( 1 + 11472777 T + 309328846570541 T^{2} + \)\(31\!\cdots\!34\)\( T^{3} + \)\(46\!\cdots\!58\)\( T^{4} + \)\(42\!\cdots\!78\)\( T^{5} + \)\(44\!\cdots\!95\)\( T^{6} + \)\(36\!\cdots\!79\)\( T^{7} + \)\(30\!\cdots\!15\)\( T^{8} + \)\(21\!\cdots\!00\)\( T^{9} + \)\(15\!\cdots\!36\)\( T^{10} + \)\(21\!\cdots\!00\)\( p^{7} T^{11} + \)\(30\!\cdots\!15\)\( p^{14} T^{12} + \)\(36\!\cdots\!79\)\( p^{21} T^{13} + \)\(44\!\cdots\!95\)\( p^{28} T^{14} + \)\(42\!\cdots\!78\)\( p^{35} T^{15} + \)\(46\!\cdots\!58\)\( p^{42} T^{16} + \)\(31\!\cdots\!34\)\( p^{49} T^{17} + 309328846570541 p^{56} T^{18} + 11472777 p^{63} T^{19} + p^{70} T^{20} \)
97 \( 1 - 3228571 T + 558143566617400 T^{2} - \)\(25\!\cdots\!15\)\( T^{3} + \)\(15\!\cdots\!75\)\( T^{4} - \)\(81\!\cdots\!02\)\( T^{5} + \)\(26\!\cdots\!96\)\( T^{6} - \)\(14\!\cdots\!86\)\( T^{7} + \)\(34\!\cdots\!33\)\( T^{8} - \)\(17\!\cdots\!25\)\( T^{9} + \)\(31\!\cdots\!64\)\( T^{10} - \)\(17\!\cdots\!25\)\( p^{7} T^{11} + \)\(34\!\cdots\!33\)\( p^{14} T^{12} - \)\(14\!\cdots\!86\)\( p^{21} T^{13} + \)\(26\!\cdots\!96\)\( p^{28} T^{14} - \)\(81\!\cdots\!02\)\( p^{35} T^{15} + \)\(15\!\cdots\!75\)\( p^{42} T^{16} - \)\(25\!\cdots\!15\)\( p^{49} T^{17} + 558143566617400 p^{56} T^{18} - 3228571 p^{63} T^{19} + p^{70} T^{20} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{20} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−4.80037475847453748240447208417, −4.76186561977956370086769896209, −4.70330233272987203415396040068, −4.59741072238162505054265542284, −4.49694805464871564065538603051, −4.10703250810770656976087154394, −4.08382039946376337335931868279, −4.01111773761644286317912745129, −3.66679164738540748871247680094, −3.57435660254497187785635750302, −3.53134722916179190378439356493, −3.44173239824496406507770334297, −3.22282454670445935353522658596, −2.99877221436532285126095867145, −2.61310674394530296847598498028, −2.44454168611612250213989416158, −2.36527736357680293643863215886, −2.29942721829494644228638490171, −1.72815867989601915891903134667, −1.69290717703264830958670281903, −1.68019321672298700179106637650, −1.67971159345017263789245250296, −1.43044142206764317448831476794, −0.995886737109739991821878512394, −0.894887741202150934763324009855, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.894887741202150934763324009855, 0.995886737109739991821878512394, 1.43044142206764317448831476794, 1.67971159345017263789245250296, 1.68019321672298700179106637650, 1.69290717703264830958670281903, 1.72815867989601915891903134667, 2.29942721829494644228638490171, 2.36527736357680293643863215886, 2.44454168611612250213989416158, 2.61310674394530296847598498028, 2.99877221436532285126095867145, 3.22282454670445935353522658596, 3.44173239824496406507770334297, 3.53134722916179190378439356493, 3.57435660254497187785635750302, 3.66679164738540748871247680094, 4.01111773761644286317912745129, 4.08382039946376337335931868279, 4.10703250810770656976087154394, 4.49694805464871564065538603051, 4.59741072238162505054265542284, 4.70330233272987203415396040068, 4.76186561977956370086769896209, 4.80037475847453748240447208417

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

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