Properties

Degree 40
Conductor $ 43^{20} $
Sign $1$
Motivic weight 6
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 242·4-s + 5.92e3·9-s + 2.61e3·11-s − 5.61e3·13-s + 2.02e4·16-s + 3.32e3·17-s − 836·23-s + 1.27e5·25-s + 1.71e4·31-s + 1.43e6·36-s − 1.41e5·41-s + 5.81e4·43-s + 6.33e5·44-s + 4.84e4·47-s + 1.16e6·49-s − 1.35e6·52-s + 4.25e5·53-s − 9.18e5·59-s + 3.14e5·64-s + 1.09e6·67-s + 8.05e5·68-s − 4.01e5·79-s + 1.71e7·81-s + 1.35e6·83-s − 2.02e5·92-s + 7.81e6·97-s + 1.54e7·99-s + ⋯
L(s)  = 1  + 3.78·4-s + 8.12·9-s + 1.96·11-s − 2.55·13-s + 4.94·16-s + 0.677·17-s − 0.0687·23-s + 8.14·25-s + 0.574·31-s + 30.7·36-s − 2.05·41-s + 0.731·43-s + 7.43·44-s + 0.466·47-s + 9.91·49-s − 9.65·52-s + 2.85·53-s − 4.47·59-s + 1.19·64-s + 3.65·67-s + 2.56·68-s − 0.814·79-s + 32.2·81-s + 2.36·83-s − 0.259·92-s + 8.56·97-s + 15.9·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{20}\right)^{s/2} \, \Gamma_{\C}(s)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(43^{20}\right)^{s/2} \, \Gamma_{\C}(s+3)^{20} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

\( d \)  =  \(40\)
\( N \)  =  \(43^{20}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(6\)
character  :  induced by $\chi_{43} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((40,\ 43^{20} ,\ ( \ : [3]^{20} ),\ 1 )\)
\(L(\frac{7}{2})\)  \(\approx\)  \(1005.11\)
\(L(\frac12)\)  \(\approx\)  \(1005.11\)
\(L(4)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 43$,\(F_p(T)\) is a polynomial of degree 40. If $p = 43$, then $F_p(T)$ is a polynomial of degree at most 39.
$p$$F_p(T)$
bad43 \( 1 - 58160 T - 3625589906 T^{2} + 9467320032272 p T^{3} + 10928962143190132 p^{2} T^{4} - 83920771107373744128 p^{3} T^{5} + \)\(16\!\cdots\!68\)\( p^{5} T^{6} + \)\(10\!\cdots\!48\)\( p^{7} T^{7} - \)\(44\!\cdots\!96\)\( p^{9} T^{8} - \)\(11\!\cdots\!12\)\( p^{11} T^{9} + \)\(29\!\cdots\!12\)\( p^{14} T^{10} - \)\(11\!\cdots\!12\)\( p^{17} T^{11} - \)\(44\!\cdots\!96\)\( p^{21} T^{12} + \)\(10\!\cdots\!48\)\( p^{25} T^{13} + \)\(16\!\cdots\!68\)\( p^{29} T^{14} - 83920771107373744128 p^{33} T^{15} + 10928962143190132 p^{38} T^{16} + 9467320032272 p^{43} T^{17} - 3625589906 p^{48} T^{18} - 58160 p^{54} T^{19} + p^{60} T^{20} \)
good2 \( 1 - 121 p T^{2} + 38293 T^{4} - 584441 p^{3} T^{6} + 7501865 p^{6} T^{8} - 2706432169 p^{4} T^{10} + 218218109207 p^{4} T^{12} - 4063258826287 p^{6} T^{14} + 35486941525393 p^{9} T^{16} - 18499154670123 p^{16} T^{18} + 37468157217543 p^{21} T^{20} - 18499154670123 p^{28} T^{22} + 35486941525393 p^{33} T^{24} - 4063258826287 p^{42} T^{26} + 218218109207 p^{52} T^{28} - 2706432169 p^{64} T^{30} + 7501865 p^{78} T^{32} - 584441 p^{87} T^{34} + 38293 p^{96} T^{36} - 121 p^{109} T^{38} + p^{120} T^{40} \)
3 \( 1 - 658 p^{2} T^{2} + 17916017 T^{4} - 36741838976 T^{6} + 19222012920194 p T^{8} - 2748035451676510 p^{3} T^{10} + 3029162578319439214 p^{3} T^{12} - \)\(98\!\cdots\!30\)\( p^{4} T^{14} + \)\(95\!\cdots\!89\)\( p^{6} T^{16} - \)\(77\!\cdots\!30\)\( p^{6} T^{18} + \)\(19\!\cdots\!98\)\( p^{7} T^{20} - \)\(77\!\cdots\!30\)\( p^{18} T^{22} + \)\(95\!\cdots\!89\)\( p^{30} T^{24} - \)\(98\!\cdots\!30\)\( p^{40} T^{26} + 3029162578319439214 p^{51} T^{28} - 2748035451676510 p^{63} T^{30} + 19222012920194 p^{73} T^{32} - 36741838976 p^{84} T^{34} + 17916017 p^{96} T^{36} - 658 p^{110} T^{38} + p^{120} T^{40} \)
5 \( 1 - 127298 T^{2} + 8822084473 T^{4} - 428558542330656 T^{6} + 3239694581934381246 p T^{8} - \)\(40\!\cdots\!74\)\( p^{3} T^{10} + \)\(42\!\cdots\!26\)\( p^{5} T^{12} - \)\(39\!\cdots\!86\)\( p^{7} T^{14} + \)\(32\!\cdots\!57\)\( p^{9} T^{16} - \)\(23\!\cdots\!22\)\( p^{11} T^{18} + \)\(62\!\cdots\!98\)\( p^{15} T^{20} - \)\(23\!\cdots\!22\)\( p^{23} T^{22} + \)\(32\!\cdots\!57\)\( p^{33} T^{24} - \)\(39\!\cdots\!86\)\( p^{43} T^{26} + \)\(42\!\cdots\!26\)\( p^{53} T^{28} - \)\(40\!\cdots\!74\)\( p^{63} T^{30} + 3239694581934381246 p^{73} T^{32} - 428558542330656 p^{84} T^{34} + 8822084473 p^{96} T^{36} - 127298 p^{108} T^{38} + p^{120} T^{40} \)
7 \( 1 - 1166168 T^{2} + 679026522922 T^{4} - 265497232908808176 T^{6} + \)\(78\!\cdots\!53\)\( T^{8} - \)\(18\!\cdots\!88\)\( T^{10} + \)\(38\!\cdots\!68\)\( T^{12} - \)\(67\!\cdots\!80\)\( T^{14} + \)\(10\!\cdots\!50\)\( T^{16} - \)\(14\!\cdots\!56\)\( T^{18} + \)\(17\!\cdots\!36\)\( T^{20} - \)\(14\!\cdots\!56\)\( p^{12} T^{22} + \)\(10\!\cdots\!50\)\( p^{24} T^{24} - \)\(67\!\cdots\!80\)\( p^{36} T^{26} + \)\(38\!\cdots\!68\)\( p^{48} T^{28} - \)\(18\!\cdots\!88\)\( p^{60} T^{30} + \)\(78\!\cdots\!53\)\( p^{72} T^{32} - 265497232908808176 p^{84} T^{34} + 679026522922 p^{96} T^{36} - 1166168 p^{108} T^{38} + p^{120} T^{40} \)
11 \( ( 1 - 1308 T + 664849 p T^{2} - 3438068668 T^{3} + 1775721407760 p T^{4} + 9588658946722524 T^{5} + 33611868196830806529 T^{6} + \)\(44\!\cdots\!20\)\( T^{7} + \)\(87\!\cdots\!39\)\( T^{8} + \)\(65\!\cdots\!76\)\( T^{9} + \)\(20\!\cdots\!36\)\( T^{10} + \)\(65\!\cdots\!76\)\( p^{6} T^{11} + \)\(87\!\cdots\!39\)\( p^{12} T^{12} + \)\(44\!\cdots\!20\)\( p^{18} T^{13} + 33611868196830806529 p^{24} T^{14} + 9588658946722524 p^{30} T^{15} + 1775721407760 p^{37} T^{16} - 3438068668 p^{42} T^{17} + 664849 p^{49} T^{18} - 1308 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
13 \( ( 1 + 2806 T + 27952549 T^{2} + 4157893422 p T^{3} + 334946569836432 T^{4} + 435014829194762046 T^{5} + \)\(24\!\cdots\!31\)\( T^{6} + \)\(20\!\cdots\!66\)\( T^{7} + \)\(10\!\cdots\!23\)\( p T^{8} + \)\(81\!\cdots\!16\)\( T^{9} + \)\(70\!\cdots\!36\)\( T^{10} + \)\(81\!\cdots\!16\)\( p^{6} T^{11} + \)\(10\!\cdots\!23\)\( p^{13} T^{12} + \)\(20\!\cdots\!66\)\( p^{18} T^{13} + \)\(24\!\cdots\!31\)\( p^{24} T^{14} + 435014829194762046 p^{30} T^{15} + 334946569836432 p^{36} T^{16} + 4157893422 p^{43} T^{17} + 27952549 p^{48} T^{18} + 2806 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
17 \( ( 1 - 1664 T + 134979316 T^{2} - 279496193526 T^{3} + 8578465814967234 T^{4} - 20710129877559965050 T^{5} + \)\(34\!\cdots\!43\)\( T^{6} - \)\(94\!\cdots\!36\)\( T^{7} + \)\(10\!\cdots\!31\)\( T^{8} - \)\(30\!\cdots\!44\)\( T^{9} + \)\(26\!\cdots\!95\)\( T^{10} - \)\(30\!\cdots\!44\)\( p^{6} T^{11} + \)\(10\!\cdots\!31\)\( p^{12} T^{12} - \)\(94\!\cdots\!36\)\( p^{18} T^{13} + \)\(34\!\cdots\!43\)\( p^{24} T^{14} - 20710129877559965050 p^{30} T^{15} + 8578465814967234 p^{36} T^{16} - 279496193526 p^{42} T^{17} + 134979316 p^{48} T^{18} - 1664 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
19 \( 1 - 426664310 T^{2} + 92809437595177093 T^{4} - \)\(13\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!10\)\( T^{8} - \)\(78\!\cdots\!18\)\( p T^{10} + \)\(11\!\cdots\!58\)\( T^{12} - \)\(83\!\cdots\!90\)\( T^{14} + \)\(52\!\cdots\!57\)\( T^{16} - \)\(28\!\cdots\!18\)\( T^{18} + \)\(14\!\cdots\!42\)\( T^{20} - \)\(28\!\cdots\!18\)\( p^{12} T^{22} + \)\(52\!\cdots\!57\)\( p^{24} T^{24} - \)\(83\!\cdots\!90\)\( p^{36} T^{26} + \)\(11\!\cdots\!58\)\( p^{48} T^{28} - \)\(78\!\cdots\!18\)\( p^{61} T^{30} + \)\(15\!\cdots\!10\)\( p^{72} T^{32} - \)\(13\!\cdots\!40\)\( p^{84} T^{34} + 92809437595177093 p^{96} T^{36} - 426664310 p^{108} T^{38} + p^{120} T^{40} \)
23 \( ( 1 + 418 T + 1035923986 T^{2} + 2202994994962 T^{3} + 42314227212656 p^{3} T^{4} + \)\(16\!\cdots\!08\)\( T^{5} + \)\(16\!\cdots\!55\)\( T^{6} + \)\(61\!\cdots\!48\)\( T^{7} + \)\(37\!\cdots\!91\)\( T^{8} + \)\(13\!\cdots\!30\)\( T^{9} + \)\(63\!\cdots\!85\)\( T^{10} + \)\(13\!\cdots\!30\)\( p^{6} T^{11} + \)\(37\!\cdots\!91\)\( p^{12} T^{12} + \)\(61\!\cdots\!48\)\( p^{18} T^{13} + \)\(16\!\cdots\!55\)\( p^{24} T^{14} + \)\(16\!\cdots\!08\)\( p^{30} T^{15} + 42314227212656 p^{39} T^{16} + 2202994994962 p^{42} T^{17} + 1035923986 p^{48} T^{18} + 418 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
29 \( 1 - 7119799082 T^{2} + 25311485826303973921 T^{4} - \)\(59\!\cdots\!28\)\( T^{6} + \)\(10\!\cdots\!78\)\( T^{8} - \)\(14\!\cdots\!26\)\( T^{10} + \)\(16\!\cdots\!90\)\( T^{12} - \)\(16\!\cdots\!14\)\( T^{14} + \)\(13\!\cdots\!41\)\( T^{16} - \)\(97\!\cdots\!06\)\( T^{18} + \)\(61\!\cdots\!10\)\( T^{20} - \)\(97\!\cdots\!06\)\( p^{12} T^{22} + \)\(13\!\cdots\!41\)\( p^{24} T^{24} - \)\(16\!\cdots\!14\)\( p^{36} T^{26} + \)\(16\!\cdots\!90\)\( p^{48} T^{28} - \)\(14\!\cdots\!26\)\( p^{60} T^{30} + \)\(10\!\cdots\!78\)\( p^{72} T^{32} - \)\(59\!\cdots\!28\)\( p^{84} T^{34} + 25311485826303973921 p^{96} T^{36} - 7119799082 p^{108} T^{38} + p^{120} T^{40} \)
31 \( ( 1 - 8558 T + 4031881282 T^{2} - 32342675026006 T^{3} + 9021132388831175984 T^{4} - \)\(81\!\cdots\!68\)\( T^{5} + \)\(13\!\cdots\!27\)\( T^{6} - \)\(14\!\cdots\!00\)\( T^{7} + \)\(16\!\cdots\!79\)\( T^{8} - \)\(17\!\cdots\!38\)\( T^{9} + \)\(16\!\cdots\!25\)\( T^{10} - \)\(17\!\cdots\!38\)\( p^{6} T^{11} + \)\(16\!\cdots\!79\)\( p^{12} T^{12} - \)\(14\!\cdots\!00\)\( p^{18} T^{13} + \)\(13\!\cdots\!27\)\( p^{24} T^{14} - \)\(81\!\cdots\!68\)\( p^{30} T^{15} + 9021132388831175984 p^{36} T^{16} - 32342675026006 p^{42} T^{17} + 4031881282 p^{48} T^{18} - 8558 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
37 \( 1 - 23384133878 T^{2} + \)\(27\!\cdots\!13\)\( T^{4} - \)\(22\!\cdots\!16\)\( T^{6} + \)\(14\!\cdots\!46\)\( T^{8} - \)\(72\!\cdots\!38\)\( T^{10} + \)\(31\!\cdots\!26\)\( T^{12} - \)\(12\!\cdots\!90\)\( T^{14} + \)\(40\!\cdots\!33\)\( T^{16} - \)\(12\!\cdots\!82\)\( T^{18} + \)\(24\!\cdots\!46\)\( p^{2} T^{20} - \)\(12\!\cdots\!82\)\( p^{12} T^{22} + \)\(40\!\cdots\!33\)\( p^{24} T^{24} - \)\(12\!\cdots\!90\)\( p^{36} T^{26} + \)\(31\!\cdots\!26\)\( p^{48} T^{28} - \)\(72\!\cdots\!38\)\( p^{60} T^{30} + \)\(14\!\cdots\!46\)\( p^{72} T^{32} - \)\(22\!\cdots\!16\)\( p^{84} T^{34} + \)\(27\!\cdots\!13\)\( p^{96} T^{36} - 23384133878 p^{108} T^{38} + p^{120} T^{40} \)
41 \( ( 1 + 70968 T + 21026049068 T^{2} + 741643261844470 T^{3} + \)\(20\!\cdots\!02\)\( T^{4} + \)\(31\!\cdots\!54\)\( T^{5} + \)\(14\!\cdots\!99\)\( T^{6} + \)\(37\!\cdots\!20\)\( T^{7} + \)\(86\!\cdots\!23\)\( T^{8} - \)\(61\!\cdots\!80\)\( T^{9} + \)\(42\!\cdots\!55\)\( T^{10} - \)\(61\!\cdots\!80\)\( p^{6} T^{11} + \)\(86\!\cdots\!23\)\( p^{12} T^{12} + \)\(37\!\cdots\!20\)\( p^{18} T^{13} + \)\(14\!\cdots\!99\)\( p^{24} T^{14} + \)\(31\!\cdots\!54\)\( p^{30} T^{15} + \)\(20\!\cdots\!02\)\( p^{36} T^{16} + 741643261844470 p^{42} T^{17} + 21026049068 p^{48} T^{18} + 70968 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
47 \( ( 1 - 24226 T + 37335944065 T^{2} + 1379124453964116 T^{3} + \)\(76\!\cdots\!82\)\( T^{4} + \)\(62\!\cdots\!26\)\( T^{5} + \)\(12\!\cdots\!02\)\( T^{6} + \)\(13\!\cdots\!22\)\( T^{7} + \)\(18\!\cdots\!89\)\( T^{8} + \)\(19\!\cdots\!06\)\( T^{9} + \)\(21\!\cdots\!62\)\( T^{10} + \)\(19\!\cdots\!06\)\( p^{6} T^{11} + \)\(18\!\cdots\!89\)\( p^{12} T^{12} + \)\(13\!\cdots\!22\)\( p^{18} T^{13} + \)\(12\!\cdots\!02\)\( p^{24} T^{14} + \)\(62\!\cdots\!26\)\( p^{30} T^{15} + \)\(76\!\cdots\!82\)\( p^{36} T^{16} + 1379124453964116 p^{42} T^{17} + 37335944065 p^{48} T^{18} - 24226 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
53 \( ( 1 - 212606 T + 117978826141 T^{2} - 27035287925250270 T^{3} + \)\(81\!\cdots\!56\)\( T^{4} - \)\(16\!\cdots\!34\)\( T^{5} + \)\(38\!\cdots\!83\)\( T^{6} - \)\(12\!\cdots\!46\)\( p T^{7} + \)\(13\!\cdots\!03\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{9} + \)\(32\!\cdots\!92\)\( T^{10} - \)\(20\!\cdots\!72\)\( p^{6} T^{11} + \)\(13\!\cdots\!03\)\( p^{12} T^{12} - \)\(12\!\cdots\!46\)\( p^{19} T^{13} + \)\(38\!\cdots\!83\)\( p^{24} T^{14} - \)\(16\!\cdots\!34\)\( p^{30} T^{15} + \)\(81\!\cdots\!56\)\( p^{36} T^{16} - 27035287925250270 p^{42} T^{17} + 117978826141 p^{48} T^{18} - 212606 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
59 \( ( 1 + 459428 T + 395999778694 T^{2} + 137746690276768556 T^{3} + \)\(68\!\cdots\!29\)\( T^{4} + \)\(19\!\cdots\!76\)\( T^{5} + \)\(71\!\cdots\!96\)\( T^{6} + \)\(16\!\cdots\!84\)\( T^{7} + \)\(50\!\cdots\!02\)\( T^{8} + \)\(10\!\cdots\!08\)\( T^{9} + \)\(24\!\cdots\!24\)\( T^{10} + \)\(10\!\cdots\!08\)\( p^{6} T^{11} + \)\(50\!\cdots\!02\)\( p^{12} T^{12} + \)\(16\!\cdots\!84\)\( p^{18} T^{13} + \)\(71\!\cdots\!96\)\( p^{24} T^{14} + \)\(19\!\cdots\!76\)\( p^{30} T^{15} + \)\(68\!\cdots\!29\)\( p^{36} T^{16} + 137746690276768556 p^{42} T^{17} + 395999778694 p^{48} T^{18} + 459428 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
61 \( 1 - 498959450864 T^{2} + \)\(12\!\cdots\!06\)\( T^{4} - \)\(20\!\cdots\!88\)\( T^{6} + \)\(26\!\cdots\!05\)\( T^{8} - \)\(28\!\cdots\!52\)\( T^{10} + \)\(25\!\cdots\!76\)\( T^{12} - \)\(19\!\cdots\!44\)\( T^{14} + \)\(13\!\cdots\!66\)\( T^{16} - \)\(81\!\cdots\!92\)\( T^{18} + \)\(44\!\cdots\!12\)\( T^{20} - \)\(81\!\cdots\!92\)\( p^{12} T^{22} + \)\(13\!\cdots\!66\)\( p^{24} T^{24} - \)\(19\!\cdots\!44\)\( p^{36} T^{26} + \)\(25\!\cdots\!76\)\( p^{48} T^{28} - \)\(28\!\cdots\!52\)\( p^{60} T^{30} + \)\(26\!\cdots\!05\)\( p^{72} T^{32} - \)\(20\!\cdots\!88\)\( p^{84} T^{34} + \)\(12\!\cdots\!06\)\( p^{96} T^{36} - 498959450864 p^{108} T^{38} + p^{120} T^{40} \)
67 \( ( 1 - 549248 T + 626059155535 T^{2} - 293286454957890652 T^{3} + \)\(18\!\cdots\!28\)\( T^{4} - \)\(79\!\cdots\!44\)\( T^{5} + \)\(37\!\cdots\!25\)\( T^{6} - \)\(13\!\cdots\!08\)\( T^{7} + \)\(51\!\cdots\!63\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} + \)\(54\!\cdots\!76\)\( T^{10} - \)\(17\!\cdots\!40\)\( p^{6} T^{11} + \)\(51\!\cdots\!63\)\( p^{12} T^{12} - \)\(13\!\cdots\!08\)\( p^{18} T^{13} + \)\(37\!\cdots\!25\)\( p^{24} T^{14} - \)\(79\!\cdots\!44\)\( p^{30} T^{15} + \)\(18\!\cdots\!28\)\( p^{36} T^{16} - 293286454957890652 p^{42} T^{17} + 626059155535 p^{48} T^{18} - 549248 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
71 \( 1 - 1099385768348 T^{2} + \)\(68\!\cdots\!30\)\( T^{4} - \)\(30\!\cdots\!96\)\( T^{6} + \)\(10\!\cdots\!65\)\( T^{8} - \)\(28\!\cdots\!20\)\( T^{10} + \)\(68\!\cdots\!88\)\( T^{12} - \)\(13\!\cdots\!72\)\( T^{14} + \)\(24\!\cdots\!18\)\( T^{16} - \)\(38\!\cdots\!40\)\( T^{18} + \)\(52\!\cdots\!28\)\( T^{20} - \)\(38\!\cdots\!40\)\( p^{12} T^{22} + \)\(24\!\cdots\!18\)\( p^{24} T^{24} - \)\(13\!\cdots\!72\)\( p^{36} T^{26} + \)\(68\!\cdots\!88\)\( p^{48} T^{28} - \)\(28\!\cdots\!20\)\( p^{60} T^{30} + \)\(10\!\cdots\!65\)\( p^{72} T^{32} - \)\(30\!\cdots\!96\)\( p^{84} T^{34} + \)\(68\!\cdots\!30\)\( p^{96} T^{36} - 1099385768348 p^{108} T^{38} + p^{120} T^{40} \)
73 \( 1 - 2077081529264 T^{2} + \)\(20\!\cdots\!78\)\( T^{4} - \)\(13\!\cdots\!76\)\( T^{6} + \)\(65\!\cdots\!45\)\( T^{8} - \)\(24\!\cdots\!96\)\( T^{10} + \)\(71\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!16\)\( T^{14} + \)\(37\!\cdots\!50\)\( T^{16} - \)\(69\!\cdots\!28\)\( T^{18} + \)\(11\!\cdots\!72\)\( T^{20} - \)\(69\!\cdots\!28\)\( p^{12} T^{22} + \)\(37\!\cdots\!50\)\( p^{24} T^{24} - \)\(17\!\cdots\!16\)\( p^{36} T^{26} + \)\(71\!\cdots\!00\)\( p^{48} T^{28} - \)\(24\!\cdots\!96\)\( p^{60} T^{30} + \)\(65\!\cdots\!45\)\( p^{72} T^{32} - \)\(13\!\cdots\!76\)\( p^{84} T^{34} + \)\(20\!\cdots\!78\)\( p^{96} T^{36} - 2077081529264 p^{108} T^{38} + p^{120} T^{40} \)
79 \( ( 1 + 200746 T + 1301669964385 T^{2} + 297228242564778316 T^{3} + \)\(87\!\cdots\!74\)\( T^{4} + \)\(19\!\cdots\!54\)\( T^{5} + \)\(40\!\cdots\!46\)\( T^{6} + \)\(79\!\cdots\!50\)\( T^{7} + \)\(13\!\cdots\!77\)\( T^{8} + \)\(24\!\cdots\!98\)\( T^{9} + \)\(37\!\cdots\!90\)\( T^{10} + \)\(24\!\cdots\!98\)\( p^{6} T^{11} + \)\(13\!\cdots\!77\)\( p^{12} T^{12} + \)\(79\!\cdots\!50\)\( p^{18} T^{13} + \)\(40\!\cdots\!46\)\( p^{24} T^{14} + \)\(19\!\cdots\!54\)\( p^{30} T^{15} + \)\(87\!\cdots\!74\)\( p^{36} T^{16} + 297228242564778316 p^{42} T^{17} + 1301669964385 p^{48} T^{18} + 200746 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
83 \( ( 1 - 677180 T + 2832058692499 T^{2} - 1643580370789010948 T^{3} + \)\(36\!\cdots\!36\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{5} + \)\(28\!\cdots\!05\)\( T^{6} - \)\(12\!\cdots\!84\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} - \)\(57\!\cdots\!92\)\( T^{9} + \)\(59\!\cdots\!84\)\( T^{10} - \)\(57\!\cdots\!92\)\( p^{6} T^{11} + \)\(15\!\cdots\!87\)\( p^{12} T^{12} - \)\(12\!\cdots\!84\)\( p^{18} T^{13} + \)\(28\!\cdots\!05\)\( p^{24} T^{14} - \)\(18\!\cdots\!20\)\( p^{30} T^{15} + \)\(36\!\cdots\!36\)\( p^{36} T^{16} - 1643580370789010948 p^{42} T^{17} + 2832058692499 p^{48} T^{18} - 677180 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
89 \( 1 - 1519004027192 T^{2} + \)\(80\!\cdots\!22\)\( T^{4} - \)\(27\!\cdots\!40\)\( T^{6} + \)\(29\!\cdots\!49\)\( T^{8} - \)\(30\!\cdots\!96\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{12} - \)\(67\!\cdots\!60\)\( T^{14} + \)\(34\!\cdots\!98\)\( T^{16} - \)\(25\!\cdots\!32\)\( T^{18} + \)\(15\!\cdots\!80\)\( T^{20} - \)\(25\!\cdots\!32\)\( p^{12} T^{22} + \)\(34\!\cdots\!98\)\( p^{24} T^{24} - \)\(67\!\cdots\!60\)\( p^{36} T^{26} + \)\(17\!\cdots\!00\)\( p^{48} T^{28} - \)\(30\!\cdots\!96\)\( p^{60} T^{30} + \)\(29\!\cdots\!49\)\( p^{72} T^{32} - \)\(27\!\cdots\!40\)\( p^{84} T^{34} + \)\(80\!\cdots\!22\)\( p^{96} T^{36} - 1519004027192 p^{108} T^{38} + p^{120} T^{40} \)
97 \( ( 1 - 3907280 T + 12196450176388 T^{2} - 26367002972894948630 T^{3} + \)\(49\!\cdots\!98\)\( T^{4} - \)\(77\!\cdots\!22\)\( T^{5} + \)\(10\!\cdots\!59\)\( T^{6} - \)\(13\!\cdots\!76\)\( T^{7} + \)\(15\!\cdots\!99\)\( T^{8} - \)\(15\!\cdots\!72\)\( T^{9} + \)\(15\!\cdots\!15\)\( T^{10} - \)\(15\!\cdots\!72\)\( p^{6} T^{11} + \)\(15\!\cdots\!99\)\( p^{12} T^{12} - \)\(13\!\cdots\!76\)\( p^{18} T^{13} + \)\(10\!\cdots\!59\)\( p^{24} T^{14} - \)\(77\!\cdots\!22\)\( p^{30} T^{15} + \)\(49\!\cdots\!98\)\( p^{36} T^{16} - 26367002972894948630 p^{42} T^{17} + 12196450176388 p^{48} T^{18} - 3907280 p^{54} T^{19} + p^{60} T^{20} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{40} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.90206634267096550531962810278, −2.72324676982548523472340457546, −2.62946233664440245287316400799, −2.60368732741431465554009586666, −2.59789374962522858603354960214, −2.42183820657512897667728424528, −2.34822331850471858502580306668, −2.24546500496106176198308224729, −2.11546369424120256196312714218, −2.10949493294047981989542394032, −1.86074671692162687021041730702, −1.84622879162317828705514349604, −1.50692198046181224653064858086, −1.42952992914123833648503340897, −1.36777951588150601660165496415, −1.35145926537174457466475865199, −1.17276957545697962098582879771, −1.14928651980621580062048321102, −0.891080355346916832721866019758, −0.879621182773044438739575679714, −0.855213602883476546646458016994, −0.799266627297453240647830662630, −0.75803919748514223617647192878, −0.24753120798455589039387095891, −0.22607126320321661333934267650, 0.22607126320321661333934267650, 0.24753120798455589039387095891, 0.75803919748514223617647192878, 0.799266627297453240647830662630, 0.855213602883476546646458016994, 0.879621182773044438739575679714, 0.891080355346916832721866019758, 1.14928651980621580062048321102, 1.17276957545697962098582879771, 1.35145926537174457466475865199, 1.36777951588150601660165496415, 1.42952992914123833648503340897, 1.50692198046181224653064858086, 1.84622879162317828705514349604, 1.86074671692162687021041730702, 2.10949493294047981989542394032, 2.11546369424120256196312714218, 2.24546500496106176198308224729, 2.34822331850471858502580306668, 2.42183820657512897667728424528, 2.59789374962522858603354960214, 2.60368732741431465554009586666, 2.62946233664440245287316400799, 2.72324676982548523472340457546, 2.90206634267096550531962810278

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

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