Properties

Degree 36
Conductor $ 2^{54} $
Sign $1$
Motivic weight 20
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 398·2-s − 1.14e5·3-s + 4.41e5·4-s + 4.54e7·6-s − 1.18e9·8-s − 1.97e10·9-s + 2.78e10·11-s − 5.04e10·12-s + 2.67e11·16-s + 4.34e12·17-s + 7.86e12·18-s − 3.60e11·19-s − 1.10e13·22-s + 1.35e14·24-s + 6.60e14·25-s + 2.48e15·27-s − 9.58e14·32-s − 3.18e15·33-s − 1.72e15·34-s − 8.72e15·36-s + 1.43e14·38-s + 1.69e16·41-s − 2.74e16·43-s + 1.23e16·44-s − 3.05e16·48-s + 5.91e17·49-s − 2.62e17·50-s + ⋯
L(s)  = 1  − 0.388·2-s − 1.93·3-s + 0.421·4-s + 0.751·6-s − 1.10·8-s − 5.66·9-s + 1.07·11-s − 0.814·12-s + 0.243·16-s + 2.15·17-s + 2.20·18-s − 0.0588·19-s − 0.417·22-s + 2.14·24-s + 6.92·25-s + 12.0·27-s − 0.851·32-s − 2.07·33-s − 0.837·34-s − 2.38·36-s + 0.0228·38-s + 1.26·41-s − 1.27·43-s + 0.452·44-s − 0.471·48-s + 7.41·49-s − 2.69·50-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{54}\right)^{s/2} \, \Gamma_{\C}(s)^{18} \, L(s)\cr =\mathstrut & \,\Lambda(21-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut &\left(2^{54}\right)^{s/2} \, \Gamma_{\C}(s+10)^{18} \, L(s)\cr =\mathstrut & \,\Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(36\)
\( N \)  =  \(2^{54}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(20\)
character  :  induced by $\chi_{8} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(36,\ 2^{54} ,\ ( \ : [10]^{18} ),\ 1 )$
$L(\frac{21}{2})$  $\approx$  $3.54420$
$L(\frac12)$  $\approx$  $3.54420$
$L(11)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 36. If $p = 2$, then $F_p$ is a polynomial of degree at most 35.
$p$$F_p$
bad2 \( 1 + 199 p T - 35411 p^{3} T^{2} + 14083337 p^{6} T^{3} + 673474055 p^{10} T^{4} + 11942422939 p^{15} T^{5} + 249191446069 p^{21} T^{6} - 703784211103 p^{28} T^{7} + 7470697516139 p^{36} T^{8} + 30675285436225 p^{45} T^{9} + 7470697516139 p^{56} T^{10} - 703784211103 p^{68} T^{11} + 249191446069 p^{81} T^{12} + 11942422939 p^{95} T^{13} + 673474055 p^{110} T^{14} + 14083337 p^{126} T^{15} - 35411 p^{143} T^{16} + 199 p^{161} T^{17} + p^{180} T^{18} \)
good3 \( ( 1 + 6346 p^{2} T + 4924040579 p T^{2} + 33948561926288 p^{3} T^{3} + 512002124596225436 p^{5} T^{4} + \)\(11\!\cdots\!56\)\( p^{8} T^{5} + \)\(41\!\cdots\!36\)\( p^{11} T^{6} + \)\(34\!\cdots\!28\)\( p^{19} T^{7} + \)\(86\!\cdots\!94\)\( p^{18} T^{8} + \)\(51\!\cdots\!16\)\( p^{22} T^{9} + \)\(86\!\cdots\!94\)\( p^{38} T^{10} + \)\(34\!\cdots\!28\)\( p^{59} T^{11} + \)\(41\!\cdots\!36\)\( p^{71} T^{12} + \)\(11\!\cdots\!56\)\( p^{88} T^{13} + 512002124596225436 p^{105} T^{14} + 33948561926288 p^{123} T^{15} + 4924040579 p^{141} T^{16} + 6346 p^{162} T^{17} + p^{180} T^{18} )^{2} \)
5 \( 1 - 132088629252234 p T^{2} + \)\(45\!\cdots\!73\)\( p T^{4} - \)\(44\!\cdots\!52\)\( p^{3} T^{6} + \)\(67\!\cdots\!24\)\( p^{6} T^{8} - \)\(87\!\cdots\!04\)\( p^{9} T^{10} + \)\(97\!\cdots\!44\)\( p^{12} T^{12} - \)\(96\!\cdots\!24\)\( p^{15} T^{14} + \)\(86\!\cdots\!22\)\( p^{18} T^{16} - \)\(13\!\cdots\!68\)\( p^{22} T^{18} + \)\(86\!\cdots\!22\)\( p^{58} T^{20} - \)\(96\!\cdots\!24\)\( p^{95} T^{22} + \)\(97\!\cdots\!44\)\( p^{132} T^{24} - \)\(87\!\cdots\!04\)\( p^{169} T^{26} + \)\(67\!\cdots\!24\)\( p^{206} T^{28} - \)\(44\!\cdots\!52\)\( p^{243} T^{30} + \)\(45\!\cdots\!73\)\( p^{281} T^{32} - 132088629252234 p^{321} T^{34} + p^{360} T^{36} \)
7 \( 1 - 1724920415944830 p^{3} T^{2} + \)\(36\!\cdots\!29\)\( p^{2} T^{4} - \)\(10\!\cdots\!40\)\( p^{3} T^{6} + \)\(24\!\cdots\!16\)\( p^{4} T^{8} - \)\(44\!\cdots\!36\)\( p^{5} T^{10} + \)\(68\!\cdots\!84\)\( p^{6} T^{12} - \)\(13\!\cdots\!72\)\( p^{8} T^{14} + \)\(47\!\cdots\!46\)\( p^{12} T^{16} - \)\(16\!\cdots\!32\)\( p^{16} T^{18} + \)\(47\!\cdots\!46\)\( p^{52} T^{20} - \)\(13\!\cdots\!72\)\( p^{88} T^{22} + \)\(68\!\cdots\!84\)\( p^{126} T^{24} - \)\(44\!\cdots\!36\)\( p^{165} T^{26} + \)\(24\!\cdots\!16\)\( p^{204} T^{28} - \)\(10\!\cdots\!40\)\( p^{243} T^{30} + \)\(36\!\cdots\!29\)\( p^{282} T^{32} - 1724920415944830 p^{323} T^{34} + p^{360} T^{36} \)
11 \( ( 1 - 13931197510 T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(13\!\cdots\!36\)\( p T^{3} + \)\(49\!\cdots\!80\)\( T^{4} - \)\(57\!\cdots\!60\)\( p T^{5} + \)\(47\!\cdots\!00\)\( p^{2} T^{6} + \)\(72\!\cdots\!00\)\( p^{3} T^{7} + \)\(34\!\cdots\!70\)\( p^{4} T^{8} + \)\(22\!\cdots\!40\)\( p^{5} T^{9} + \)\(34\!\cdots\!70\)\( p^{24} T^{10} + \)\(72\!\cdots\!00\)\( p^{43} T^{11} + \)\(47\!\cdots\!00\)\( p^{62} T^{12} - \)\(57\!\cdots\!60\)\( p^{81} T^{13} + \)\(49\!\cdots\!80\)\( p^{100} T^{14} - \)\(13\!\cdots\!36\)\( p^{121} T^{15} + \)\(30\!\cdots\!45\)\( p^{140} T^{16} - 13931197510 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
13 \( 1 - \)\(14\!\cdots\!90\)\( T^{2} + \)\(11\!\cdots\!01\)\( T^{4} - \)\(66\!\cdots\!40\)\( T^{6} + \)\(22\!\cdots\!12\)\( p T^{8} - \)\(47\!\cdots\!96\)\( p^{3} T^{10} + \)\(87\!\cdots\!12\)\( p^{5} T^{12} - \)\(13\!\cdots\!56\)\( p^{7} T^{14} + \)\(18\!\cdots\!62\)\( p^{9} T^{16} - \)\(22\!\cdots\!76\)\( p^{11} T^{18} + \)\(18\!\cdots\!62\)\( p^{49} T^{20} - \)\(13\!\cdots\!56\)\( p^{87} T^{22} + \)\(87\!\cdots\!12\)\( p^{125} T^{24} - \)\(47\!\cdots\!96\)\( p^{163} T^{26} + \)\(22\!\cdots\!12\)\( p^{201} T^{28} - \)\(66\!\cdots\!40\)\( p^{240} T^{30} + \)\(11\!\cdots\!01\)\( p^{280} T^{32} - \)\(14\!\cdots\!90\)\( p^{320} T^{34} + p^{360} T^{36} \)
17 \( ( 1 - 2171962569586 T + \)\(68\!\cdots\!13\)\( p^{2} T^{2} - \)\(14\!\cdots\!36\)\( p^{2} T^{3} + \)\(41\!\cdots\!36\)\( p^{3} T^{4} - \)\(49\!\cdots\!44\)\( p^{4} T^{5} + \)\(10\!\cdots\!96\)\( p^{5} T^{6} - \)\(11\!\cdots\!56\)\( p^{6} T^{7} + \)\(18\!\cdots\!02\)\( p^{7} T^{8} - \)\(18\!\cdots\!76\)\( p^{8} T^{9} + \)\(18\!\cdots\!02\)\( p^{27} T^{10} - \)\(11\!\cdots\!56\)\( p^{46} T^{11} + \)\(10\!\cdots\!96\)\( p^{65} T^{12} - \)\(49\!\cdots\!44\)\( p^{84} T^{13} + \)\(41\!\cdots\!36\)\( p^{103} T^{14} - \)\(14\!\cdots\!36\)\( p^{122} T^{15} + \)\(68\!\cdots\!13\)\( p^{142} T^{16} - 2171962569586 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
19 \( ( 1 + 9491622462 p T + \)\(47\!\cdots\!17\)\( p^{2} T^{2} + \)\(31\!\cdots\!08\)\( p^{3} T^{3} + \)\(12\!\cdots\!24\)\( p^{4} T^{4} + \)\(10\!\cdots\!92\)\( p^{5} T^{5} + \)\(22\!\cdots\!20\)\( p^{6} T^{6} + \)\(18\!\cdots\!80\)\( p^{7} T^{7} + \)\(16\!\cdots\!42\)\( p^{9} T^{8} + \)\(23\!\cdots\!76\)\( p^{9} T^{9} + \)\(16\!\cdots\!42\)\( p^{29} T^{10} + \)\(18\!\cdots\!80\)\( p^{47} T^{11} + \)\(22\!\cdots\!20\)\( p^{66} T^{12} + \)\(10\!\cdots\!92\)\( p^{85} T^{13} + \)\(12\!\cdots\!24\)\( p^{104} T^{14} + \)\(31\!\cdots\!08\)\( p^{123} T^{15} + \)\(47\!\cdots\!17\)\( p^{142} T^{16} + 9491622462 p^{161} T^{17} + p^{180} T^{18} )^{2} \)
23 \( 1 - \)\(18\!\cdots\!30\)\( T^{2} + \)\(71\!\cdots\!87\)\( p T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(45\!\cdots\!16\)\( T^{8} - \)\(16\!\cdots\!12\)\( T^{10} + \)\(49\!\cdots\!36\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{14} + \)\(26\!\cdots\!46\)\( T^{16} - \)\(49\!\cdots\!32\)\( T^{18} + \)\(26\!\cdots\!46\)\( p^{40} T^{20} - \)\(12\!\cdots\!32\)\( p^{80} T^{22} + \)\(49\!\cdots\!36\)\( p^{120} T^{24} - \)\(16\!\cdots\!12\)\( p^{160} T^{26} + \)\(45\!\cdots\!16\)\( p^{200} T^{28} - \)\(10\!\cdots\!60\)\( p^{240} T^{30} + \)\(71\!\cdots\!87\)\( p^{281} T^{32} - \)\(18\!\cdots\!30\)\( p^{320} T^{34} + p^{360} T^{36} \)
29 \( 1 - \)\(16\!\cdots\!78\)\( T^{2} + \)\(13\!\cdots\!13\)\( T^{4} - \)\(75\!\cdots\!36\)\( T^{6} + \)\(30\!\cdots\!00\)\( T^{8} - \)\(99\!\cdots\!88\)\( T^{10} + \)\(91\!\cdots\!36\)\( p T^{12} - \)\(61\!\cdots\!04\)\( T^{14} + \)\(12\!\cdots\!98\)\( T^{16} - \)\(23\!\cdots\!00\)\( T^{18} + \)\(12\!\cdots\!98\)\( p^{40} T^{20} - \)\(61\!\cdots\!04\)\( p^{80} T^{22} + \)\(91\!\cdots\!36\)\( p^{121} T^{24} - \)\(99\!\cdots\!88\)\( p^{160} T^{26} + \)\(30\!\cdots\!00\)\( p^{200} T^{28} - \)\(75\!\cdots\!36\)\( p^{240} T^{30} + \)\(13\!\cdots\!13\)\( p^{280} T^{32} - \)\(16\!\cdots\!78\)\( p^{320} T^{34} + p^{360} T^{36} \)
31 \( 1 - \)\(58\!\cdots\!78\)\( T^{2} + \)\(17\!\cdots\!13\)\( T^{4} - \)\(35\!\cdots\!16\)\( T^{6} + \)\(54\!\cdots\!60\)\( T^{8} - \)\(68\!\cdots\!68\)\( T^{10} + \)\(72\!\cdots\!44\)\( T^{12} - \)\(66\!\cdots\!04\)\( T^{14} + \)\(53\!\cdots\!58\)\( T^{16} - \)\(38\!\cdots\!20\)\( T^{18} + \)\(53\!\cdots\!58\)\( p^{40} T^{20} - \)\(66\!\cdots\!04\)\( p^{80} T^{22} + \)\(72\!\cdots\!44\)\( p^{120} T^{24} - \)\(68\!\cdots\!68\)\( p^{160} T^{26} + \)\(54\!\cdots\!60\)\( p^{200} T^{28} - \)\(35\!\cdots\!16\)\( p^{240} T^{30} + \)\(17\!\cdots\!13\)\( p^{280} T^{32} - \)\(58\!\cdots\!78\)\( p^{320} T^{34} + p^{360} T^{36} \)
37 \( 1 - \)\(21\!\cdots\!90\)\( T^{2} + \)\(23\!\cdots\!01\)\( T^{4} - \)\(16\!\cdots\!00\)\( T^{6} + \)\(90\!\cdots\!96\)\( T^{8} - \)\(39\!\cdots\!32\)\( T^{10} + \)\(14\!\cdots\!96\)\( T^{12} - \)\(46\!\cdots\!32\)\( T^{14} + \)\(13\!\cdots\!06\)\( T^{16} - \)\(32\!\cdots\!92\)\( T^{18} + \)\(13\!\cdots\!06\)\( p^{40} T^{20} - \)\(46\!\cdots\!32\)\( p^{80} T^{22} + \)\(14\!\cdots\!96\)\( p^{120} T^{24} - \)\(39\!\cdots\!32\)\( p^{160} T^{26} + \)\(90\!\cdots\!96\)\( p^{200} T^{28} - \)\(16\!\cdots\!00\)\( p^{240} T^{30} + \)\(23\!\cdots\!01\)\( p^{280} T^{32} - \)\(21\!\cdots\!90\)\( p^{320} T^{34} + p^{360} T^{36} \)
41 \( ( 1 - 8495414878199410 T + \)\(10\!\cdots\!25\)\( T^{2} - \)\(91\!\cdots\!76\)\( T^{3} + \)\(55\!\cdots\!00\)\( T^{4} - \)\(45\!\cdots\!20\)\( T^{5} + \)\(19\!\cdots\!20\)\( T^{6} - \)\(13\!\cdots\!40\)\( T^{7} + \)\(46\!\cdots\!70\)\( T^{8} - \)\(29\!\cdots\!40\)\( T^{9} + \)\(46\!\cdots\!70\)\( p^{20} T^{10} - \)\(13\!\cdots\!40\)\( p^{40} T^{11} + \)\(19\!\cdots\!20\)\( p^{60} T^{12} - \)\(45\!\cdots\!20\)\( p^{80} T^{13} + \)\(55\!\cdots\!00\)\( p^{100} T^{14} - \)\(91\!\cdots\!76\)\( p^{120} T^{15} + \)\(10\!\cdots\!25\)\( p^{140} T^{16} - 8495414878199410 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
43 \( ( 1 + 13749233178376698 T + \)\(23\!\cdots\!17\)\( T^{2} + \)\(14\!\cdots\!08\)\( T^{3} + \)\(24\!\cdots\!00\)\( T^{4} - \)\(86\!\cdots\!08\)\( T^{5} + \)\(17\!\cdots\!68\)\( T^{6} - \)\(62\!\cdots\!08\)\( T^{7} + \)\(10\!\cdots\!14\)\( T^{8} - \)\(39\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!14\)\( p^{20} T^{10} - \)\(62\!\cdots\!08\)\( p^{40} T^{11} + \)\(17\!\cdots\!68\)\( p^{60} T^{12} - \)\(86\!\cdots\!08\)\( p^{80} T^{13} + \)\(24\!\cdots\!00\)\( p^{100} T^{14} + \)\(14\!\cdots\!08\)\( p^{120} T^{15} + \)\(23\!\cdots\!17\)\( p^{140} T^{16} + 13749233178376698 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
47 \( 1 - \)\(28\!\cdots\!30\)\( T^{2} + \)\(41\!\cdots\!01\)\( T^{4} - \)\(41\!\cdots\!40\)\( T^{6} + \)\(31\!\cdots\!36\)\( T^{8} - \)\(18\!\cdots\!72\)\( T^{10} + \)\(90\!\cdots\!76\)\( T^{12} - \)\(37\!\cdots\!72\)\( T^{14} + \)\(13\!\cdots\!86\)\( T^{16} - \)\(38\!\cdots\!72\)\( T^{18} + \)\(13\!\cdots\!86\)\( p^{40} T^{20} - \)\(37\!\cdots\!72\)\( p^{80} T^{22} + \)\(90\!\cdots\!76\)\( p^{120} T^{24} - \)\(18\!\cdots\!72\)\( p^{160} T^{26} + \)\(31\!\cdots\!36\)\( p^{200} T^{28} - \)\(41\!\cdots\!40\)\( p^{240} T^{30} + \)\(41\!\cdots\!01\)\( p^{280} T^{32} - \)\(28\!\cdots\!30\)\( p^{320} T^{34} + p^{360} T^{36} \)
53 \( 1 - \)\(12\!\cdots\!70\)\( T^{2} + \)\(94\!\cdots\!21\)\( T^{4} - \)\(54\!\cdots\!00\)\( T^{6} + \)\(25\!\cdots\!96\)\( T^{8} - \)\(10\!\cdots\!32\)\( T^{10} + \)\(36\!\cdots\!76\)\( T^{12} - \)\(12\!\cdots\!52\)\( T^{14} + \)\(38\!\cdots\!06\)\( T^{16} - \)\(12\!\cdots\!92\)\( T^{18} + \)\(38\!\cdots\!06\)\( p^{40} T^{20} - \)\(12\!\cdots\!52\)\( p^{80} T^{22} + \)\(36\!\cdots\!76\)\( p^{120} T^{24} - \)\(10\!\cdots\!32\)\( p^{160} T^{26} + \)\(25\!\cdots\!96\)\( p^{200} T^{28} - \)\(54\!\cdots\!00\)\( p^{240} T^{30} + \)\(94\!\cdots\!21\)\( p^{280} T^{32} - \)\(12\!\cdots\!70\)\( p^{320} T^{34} + p^{360} T^{36} \)
59 \( ( 1 - 46556097350183110 T + \)\(19\!\cdots\!61\)\( T^{2} - \)\(56\!\cdots\!40\)\( T^{3} + \)\(17\!\cdots\!16\)\( T^{4} - \)\(31\!\cdots\!32\)\( T^{5} + \)\(97\!\cdots\!76\)\( T^{6} - \)\(11\!\cdots\!52\)\( T^{7} + \)\(36\!\cdots\!46\)\( T^{8} - \)\(30\!\cdots\!32\)\( T^{9} + \)\(36\!\cdots\!46\)\( p^{20} T^{10} - \)\(11\!\cdots\!52\)\( p^{40} T^{11} + \)\(97\!\cdots\!76\)\( p^{60} T^{12} - \)\(31\!\cdots\!32\)\( p^{80} T^{13} + \)\(17\!\cdots\!16\)\( p^{100} T^{14} - \)\(56\!\cdots\!40\)\( p^{120} T^{15} + \)\(19\!\cdots\!61\)\( p^{140} T^{16} - 46556097350183110 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
61 \( 1 - \)\(57\!\cdots\!98\)\( T^{2} + \)\(16\!\cdots\!93\)\( T^{4} - \)\(29\!\cdots\!36\)\( T^{6} + \)\(40\!\cdots\!80\)\( T^{8} - \)\(42\!\cdots\!88\)\( T^{10} + \)\(37\!\cdots\!64\)\( T^{12} - \)\(27\!\cdots\!84\)\( T^{14} + \)\(17\!\cdots\!98\)\( T^{16} - \)\(94\!\cdots\!60\)\( T^{18} + \)\(17\!\cdots\!98\)\( p^{40} T^{20} - \)\(27\!\cdots\!84\)\( p^{80} T^{22} + \)\(37\!\cdots\!64\)\( p^{120} T^{24} - \)\(42\!\cdots\!88\)\( p^{160} T^{26} + \)\(40\!\cdots\!80\)\( p^{200} T^{28} - \)\(29\!\cdots\!36\)\( p^{240} T^{30} + \)\(16\!\cdots\!93\)\( p^{280} T^{32} - \)\(57\!\cdots\!98\)\( p^{320} T^{34} + p^{360} T^{36} \)
67 \( ( 1 + 726322821037167834 T + \)\(16\!\cdots\!97\)\( T^{2} + \)\(21\!\cdots\!76\)\( T^{3} + \)\(13\!\cdots\!28\)\( T^{4} + \)\(21\!\cdots\!56\)\( T^{5} + \)\(82\!\cdots\!52\)\( T^{6} + \)\(12\!\cdots\!76\)\( T^{7} + \)\(37\!\cdots\!66\)\( T^{8} + \)\(46\!\cdots\!24\)\( T^{9} + \)\(37\!\cdots\!66\)\( p^{20} T^{10} + \)\(12\!\cdots\!76\)\( p^{40} T^{11} + \)\(82\!\cdots\!52\)\( p^{60} T^{12} + \)\(21\!\cdots\!56\)\( p^{80} T^{13} + \)\(13\!\cdots\!28\)\( p^{100} T^{14} + \)\(21\!\cdots\!76\)\( p^{120} T^{15} + \)\(16\!\cdots\!97\)\( p^{140} T^{16} + 726322821037167834 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
71 \( 1 - \)\(12\!\cdots\!78\)\( T^{2} + \)\(80\!\cdots\!53\)\( T^{4} - \)\(33\!\cdots\!56\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{8} - \)\(24\!\cdots\!08\)\( T^{10} + \)\(48\!\cdots\!84\)\( T^{12} - \)\(77\!\cdots\!84\)\( T^{14} + \)\(10\!\cdots\!38\)\( T^{16} - \)\(12\!\cdots\!20\)\( T^{18} + \)\(10\!\cdots\!38\)\( p^{40} T^{20} - \)\(77\!\cdots\!84\)\( p^{80} T^{22} + \)\(48\!\cdots\!84\)\( p^{120} T^{24} - \)\(24\!\cdots\!08\)\( p^{160} T^{26} + \)\(10\!\cdots\!60\)\( p^{200} T^{28} - \)\(33\!\cdots\!56\)\( p^{240} T^{30} + \)\(80\!\cdots\!53\)\( p^{280} T^{32} - \)\(12\!\cdots\!78\)\( p^{320} T^{34} + p^{360} T^{36} \)
73 \( ( 1 - 4941410656931587986 T + \)\(92\!\cdots\!77\)\( T^{2} - \)\(41\!\cdots\!64\)\( T^{3} + \)\(39\!\cdots\!88\)\( T^{4} - \)\(17\!\cdots\!04\)\( T^{5} + \)\(10\!\cdots\!72\)\( T^{6} - \)\(48\!\cdots\!24\)\( T^{7} + \)\(21\!\cdots\!46\)\( T^{8} - \)\(10\!\cdots\!96\)\( T^{9} + \)\(21\!\cdots\!46\)\( p^{20} T^{10} - \)\(48\!\cdots\!24\)\( p^{40} T^{11} + \)\(10\!\cdots\!72\)\( p^{60} T^{12} - \)\(17\!\cdots\!04\)\( p^{80} T^{13} + \)\(39\!\cdots\!88\)\( p^{100} T^{14} - \)\(41\!\cdots\!64\)\( p^{120} T^{15} + \)\(92\!\cdots\!77\)\( p^{140} T^{16} - 4941410656931587986 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
79 \( 1 - \)\(91\!\cdots\!18\)\( T^{2} + \)\(41\!\cdots\!73\)\( T^{4} - \)\(12\!\cdots\!16\)\( T^{6} + \)\(26\!\cdots\!20\)\( T^{8} - \)\(46\!\cdots\!68\)\( T^{10} + \)\(66\!\cdots\!84\)\( T^{12} - \)\(81\!\cdots\!64\)\( T^{14} + \)\(87\!\cdots\!58\)\( T^{16} - \)\(83\!\cdots\!40\)\( T^{18} + \)\(87\!\cdots\!58\)\( p^{40} T^{20} - \)\(81\!\cdots\!64\)\( p^{80} T^{22} + \)\(66\!\cdots\!84\)\( p^{120} T^{24} - \)\(46\!\cdots\!68\)\( p^{160} T^{26} + \)\(26\!\cdots\!20\)\( p^{200} T^{28} - \)\(12\!\cdots\!16\)\( p^{240} T^{30} + \)\(41\!\cdots\!73\)\( p^{280} T^{32} - \)\(91\!\cdots\!18\)\( p^{320} T^{34} + p^{360} T^{36} \)
83 \( ( 1 + 23212764984134278874 T + \)\(98\!\cdots\!17\)\( T^{2} + \)\(92\!\cdots\!96\)\( T^{3} + \)\(33\!\cdots\!28\)\( T^{4} + \)\(13\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!04\)\( p T^{6} + \)\(36\!\cdots\!56\)\( T^{7} + \)\(32\!\cdots\!86\)\( T^{8} + \)\(87\!\cdots\!04\)\( T^{9} + \)\(32\!\cdots\!86\)\( p^{20} T^{10} + \)\(36\!\cdots\!56\)\( p^{40} T^{11} + \)\(12\!\cdots\!04\)\( p^{61} T^{12} + \)\(13\!\cdots\!76\)\( p^{80} T^{13} + \)\(33\!\cdots\!28\)\( p^{100} T^{14} + \)\(92\!\cdots\!96\)\( p^{120} T^{15} + \)\(98\!\cdots\!17\)\( p^{140} T^{16} + 23212764984134278874 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
89 \( ( 1 + 14037585336125420078 T + \)\(51\!\cdots\!57\)\( T^{2} + \)\(80\!\cdots\!72\)\( T^{3} + \)\(13\!\cdots\!64\)\( T^{4} + \)\(21\!\cdots\!88\)\( T^{5} + \)\(24\!\cdots\!40\)\( T^{6} + \)\(37\!\cdots\!80\)\( T^{7} + \)\(32\!\cdots\!78\)\( T^{8} + \)\(43\!\cdots\!44\)\( T^{9} + \)\(32\!\cdots\!78\)\( p^{20} T^{10} + \)\(37\!\cdots\!80\)\( p^{40} T^{11} + \)\(24\!\cdots\!40\)\( p^{60} T^{12} + \)\(21\!\cdots\!88\)\( p^{80} T^{13} + \)\(13\!\cdots\!64\)\( p^{100} T^{14} + \)\(80\!\cdots\!72\)\( p^{120} T^{15} + \)\(51\!\cdots\!57\)\( p^{140} T^{16} + 14037585336125420078 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
97 \( ( 1 - 63638322586683325362 T + \)\(25\!\cdots\!97\)\( T^{2} - \)\(16\!\cdots\!32\)\( T^{3} + \)\(35\!\cdots\!00\)\( T^{4} - \)\(23\!\cdots\!88\)\( T^{5} + \)\(32\!\cdots\!88\)\( T^{6} - \)\(20\!\cdots\!68\)\( T^{7} + \)\(23\!\cdots\!14\)\( T^{8} - \)\(13\!\cdots\!00\)\( T^{9} + \)\(23\!\cdots\!14\)\( p^{20} T^{10} - \)\(20\!\cdots\!68\)\( p^{40} T^{11} + \)\(32\!\cdots\!88\)\( p^{60} T^{12} - \)\(23\!\cdots\!88\)\( p^{80} T^{13} + \)\(35\!\cdots\!00\)\( p^{100} T^{14} - \)\(16\!\cdots\!32\)\( p^{120} T^{15} + \)\(25\!\cdots\!97\)\( p^{140} T^{16} - 63638322586683325362 p^{160} T^{17} + p^{180} T^{18} )^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{36} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−2.83691087912140014563343018124, −2.71115050930887708228136505817, −2.56326813991890541643262161232, −2.53498933115058389980863431139, −2.49784556041202800474571492228, −2.48616065117148326012596020780, −2.25898271294691408625748531527, −2.18530884784956870021133266016, −1.89266412092521303620052162985, −1.86696626341114144736833152318, −1.39973290110716683551363838174, −1.37926410338343589501915301311, −1.29269493636689429663224909885, −1.25987562334497532641399650103, −1.13687142724146467263033837589, −1.10756194422560305964128884921, −0.865791374022349318508394261388, −0.803367619662063441917762232673, −0.70988937495203352185632857709, −0.56480692835002547254947116101, −0.40395241721850415079895435193, −0.32023787401404535968313928193, −0.24415449313720473725729478692, −0.23457097092372126837501991018, −0.21647337150744613969396672561, 0.21647337150744613969396672561, 0.23457097092372126837501991018, 0.24415449313720473725729478692, 0.32023787401404535968313928193, 0.40395241721850415079895435193, 0.56480692835002547254947116101, 0.70988937495203352185632857709, 0.803367619662063441917762232673, 0.865791374022349318508394261388, 1.10756194422560305964128884921, 1.13687142724146467263033837589, 1.25987562334497532641399650103, 1.29269493636689429663224909885, 1.37926410338343589501915301311, 1.39973290110716683551363838174, 1.86696626341114144736833152318, 1.89266412092521303620052162985, 2.18530884784956870021133266016, 2.25898271294691408625748531527, 2.48616065117148326012596020780, 2.49784556041202800474571492228, 2.53498933115058389980863431139, 2.56326813991890541643262161232, 2.71115050930887708228136505817, 2.83691087912140014563343018124

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

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