Properties

Label 32-252e16-1.1-c11e16-0-0
Degree $32$
Conductor $2.645\times 10^{38}$
Sign $1$
Analytic cond. $3.90204\times 10^{36}$
Root an. cond. $13.9148$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.15e3·5-s + 5.05e4·7-s + 2.22e5·11-s + 2.70e6·13-s − 5.11e6·17-s + 6.91e6·19-s + 5.13e7·23-s + 1.01e8·25-s − 1.18e8·29-s + 1.64e8·31-s + 1.08e8·35-s + 7.56e7·37-s + 1.81e9·41-s + 1.07e7·43-s + 1.03e9·47-s + 3.33e9·49-s + 6.65e8·53-s + 4.80e8·55-s − 1.04e9·59-s − 1.43e10·61-s + 5.82e9·65-s − 3.33e10·67-s − 6.58e10·71-s + 1.77e10·73-s + 1.12e10·77-s − 2.66e10·79-s + 2.10e11·83-s + ⋯
L(s)  = 1  + 0.308·5-s + 1.13·7-s + 0.417·11-s + 2.01·13-s − 0.873·17-s + 0.640·19-s + 1.66·23-s + 2.08·25-s − 1.07·29-s + 1.03·31-s + 0.350·35-s + 0.179·37-s + 2.44·41-s + 0.0111·43-s + 0.657·47-s + 1.68·49-s + 0.218·53-s + 0.128·55-s − 0.189·59-s − 2.18·61-s + 0.623·65-s − 3.01·67-s − 4.33·71-s + 0.999·73-s + 0.473·77-s − 0.973·79-s + 5.86·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{32} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(2^{32} \cdot 3^{32} \cdot 7^{16}\)
Sign: $1$
Analytic conductor: \(3.90204\times 10^{36}\)
Root analytic conductor: \(13.9148\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{252} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((32,\ 2^{32} \cdot 3^{32} \cdot 7^{16} ,\ ( \ : [11/2]^{16} ),\ 1 )\)

Particular Values

\(L(6)\) \(\approx\) \(19.56358135\)
\(L(\frac12)\) \(\approx\) \(19.56358135\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - 7216 p T - 786017352 T^{2} + 5677955013848 p T^{3} + 13189568370833794 p^{3} T^{4} - 1042456804253472128 p^{6} T^{5} - 39498882674071568384 p^{10} T^{6} + \)\(48\!\cdots\!52\)\( p^{14} T^{7} + \)\(77\!\cdots\!88\)\( p^{19} T^{8} + \)\(48\!\cdots\!52\)\( p^{25} T^{9} - 39498882674071568384 p^{32} T^{10} - 1042456804253472128 p^{39} T^{11} + 13189568370833794 p^{47} T^{12} + 5677955013848 p^{56} T^{13} - 786017352 p^{66} T^{14} - 7216 p^{78} T^{15} + p^{88} T^{16} \)
good5 \( 1 - 2156 T - 97260146 T^{2} + 797791957768 T^{3} + 3239617078952719 T^{4} - 15790640217045297432 p T^{5} + \)\(15\!\cdots\!78\)\( p^{2} T^{6} + \)\(32\!\cdots\!88\)\( p^{3} T^{7} - \)\(74\!\cdots\!19\)\( p^{4} T^{8} + \)\(14\!\cdots\!28\)\( p^{5} T^{9} + \)\(16\!\cdots\!56\)\( p^{6} T^{10} - \)\(22\!\cdots\!48\)\( p^{7} T^{11} - \)\(74\!\cdots\!94\)\( p^{8} T^{12} + \)\(60\!\cdots\!32\)\( p^{9} T^{13} - \)\(53\!\cdots\!72\)\( p^{10} T^{14} - \)\(51\!\cdots\!48\)\( p^{11} T^{15} + \)\(18\!\cdots\!34\)\( p^{12} T^{16} - \)\(51\!\cdots\!48\)\( p^{22} T^{17} - \)\(53\!\cdots\!72\)\( p^{32} T^{18} + \)\(60\!\cdots\!32\)\( p^{42} T^{19} - \)\(74\!\cdots\!94\)\( p^{52} T^{20} - \)\(22\!\cdots\!48\)\( p^{62} T^{21} + \)\(16\!\cdots\!56\)\( p^{72} T^{22} + \)\(14\!\cdots\!28\)\( p^{82} T^{23} - \)\(74\!\cdots\!19\)\( p^{92} T^{24} + \)\(32\!\cdots\!88\)\( p^{102} T^{25} + \)\(15\!\cdots\!78\)\( p^{112} T^{26} - 15790640217045297432 p^{122} T^{27} + 3239617078952719 p^{132} T^{28} + 797791957768 p^{143} T^{29} - 97260146 p^{154} T^{30} - 2156 p^{165} T^{31} + p^{176} T^{32} \)
11 \( 1 - 222796 T - 1067062769030 T^{2} + 46297508992935896 T^{3} + \)\(59\!\cdots\!71\)\( T^{4} + \)\(65\!\cdots\!72\)\( T^{5} - \)\(21\!\cdots\!26\)\( T^{6} - \)\(52\!\cdots\!92\)\( T^{7} + \)\(44\!\cdots\!63\)\( p T^{8} + \)\(21\!\cdots\!28\)\( T^{9} - \)\(35\!\cdots\!72\)\( T^{10} - \)\(59\!\cdots\!00\)\( T^{11} - \)\(28\!\cdots\!82\)\( T^{12} + \)\(12\!\cdots\!24\)\( T^{13} + \)\(17\!\cdots\!48\)\( T^{14} - \)\(13\!\cdots\!60\)\( T^{15} - \)\(58\!\cdots\!26\)\( T^{16} - \)\(13\!\cdots\!60\)\( p^{11} T^{17} + \)\(17\!\cdots\!48\)\( p^{22} T^{18} + \)\(12\!\cdots\!24\)\( p^{33} T^{19} - \)\(28\!\cdots\!82\)\( p^{44} T^{20} - \)\(59\!\cdots\!00\)\( p^{55} T^{21} - \)\(35\!\cdots\!72\)\( p^{66} T^{22} + \)\(21\!\cdots\!28\)\( p^{77} T^{23} + \)\(44\!\cdots\!63\)\( p^{89} T^{24} - \)\(52\!\cdots\!92\)\( p^{99} T^{25} - \)\(21\!\cdots\!26\)\( p^{110} T^{26} + \)\(65\!\cdots\!72\)\( p^{121} T^{27} + \)\(59\!\cdots\!71\)\( p^{132} T^{28} + 46297508992935896 p^{143} T^{29} - 1067062769030 p^{154} T^{30} - 222796 p^{165} T^{31} + p^{176} T^{32} \)
13 \( ( 1 - 1351588 T + 4451725590750 T^{2} - 7158510204689576592 T^{3} + \)\(14\!\cdots\!57\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{5} + \)\(30\!\cdots\!90\)\( T^{6} - \)\(36\!\cdots\!28\)\( T^{7} + \)\(58\!\cdots\!72\)\( T^{8} - \)\(36\!\cdots\!28\)\( p^{11} T^{9} + \)\(30\!\cdots\!90\)\( p^{22} T^{10} - \)\(18\!\cdots\!20\)\( p^{33} T^{11} + \)\(14\!\cdots\!57\)\( p^{44} T^{12} - 7158510204689576592 p^{55} T^{13} + 4451725590750 p^{66} T^{14} - 1351588 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
17 \( 1 + 5114600 T - 128769314191496 T^{2} - \)\(73\!\cdots\!60\)\( T^{3} + \)\(72\!\cdots\!32\)\( T^{4} + \)\(46\!\cdots\!00\)\( T^{5} - \)\(29\!\cdots\!36\)\( T^{6} - \)\(21\!\cdots\!00\)\( T^{7} + \)\(11\!\cdots\!34\)\( T^{8} + \)\(85\!\cdots\!40\)\( T^{9} - \)\(39\!\cdots\!96\)\( T^{10} - \)\(16\!\cdots\!20\)\( p T^{11} + \)\(12\!\cdots\!56\)\( T^{12} + \)\(65\!\cdots\!40\)\( T^{13} - \)\(37\!\cdots\!32\)\( T^{14} - \)\(44\!\cdots\!20\)\( p T^{15} + \)\(40\!\cdots\!79\)\( p^{2} T^{16} - \)\(44\!\cdots\!20\)\( p^{12} T^{17} - \)\(37\!\cdots\!32\)\( p^{22} T^{18} + \)\(65\!\cdots\!40\)\( p^{33} T^{19} + \)\(12\!\cdots\!56\)\( p^{44} T^{20} - \)\(16\!\cdots\!20\)\( p^{56} T^{21} - \)\(39\!\cdots\!96\)\( p^{66} T^{22} + \)\(85\!\cdots\!40\)\( p^{77} T^{23} + \)\(11\!\cdots\!34\)\( p^{88} T^{24} - \)\(21\!\cdots\!00\)\( p^{99} T^{25} - \)\(29\!\cdots\!36\)\( p^{110} T^{26} + \)\(46\!\cdots\!00\)\( p^{121} T^{27} + \)\(72\!\cdots\!32\)\( p^{132} T^{28} - \)\(73\!\cdots\!60\)\( p^{143} T^{29} - 128769314191496 p^{154} T^{30} + 5114600 p^{165} T^{31} + p^{176} T^{32} \)
19 \( 1 - 6910556 T - 253548316318958 T^{2} + \)\(22\!\cdots\!20\)\( T^{3} + \)\(29\!\cdots\!27\)\( T^{4} - \)\(52\!\cdots\!52\)\( T^{5} - \)\(20\!\cdots\!86\)\( T^{6} - \)\(72\!\cdots\!88\)\( T^{7} + \)\(36\!\cdots\!97\)\( T^{8} + \)\(15\!\cdots\!72\)\( T^{9} - \)\(41\!\cdots\!48\)\( T^{10} - \)\(79\!\cdots\!72\)\( p T^{11} - \)\(50\!\cdots\!74\)\( T^{12} + \)\(94\!\cdots\!84\)\( T^{13} + \)\(22\!\cdots\!80\)\( T^{14} - \)\(17\!\cdots\!52\)\( p T^{15} - \)\(96\!\cdots\!34\)\( p^{2} T^{16} - \)\(17\!\cdots\!52\)\( p^{12} T^{17} + \)\(22\!\cdots\!80\)\( p^{22} T^{18} + \)\(94\!\cdots\!84\)\( p^{33} T^{19} - \)\(50\!\cdots\!74\)\( p^{44} T^{20} - \)\(79\!\cdots\!72\)\( p^{56} T^{21} - \)\(41\!\cdots\!48\)\( p^{66} T^{22} + \)\(15\!\cdots\!72\)\( p^{77} T^{23} + \)\(36\!\cdots\!97\)\( p^{88} T^{24} - \)\(72\!\cdots\!88\)\( p^{99} T^{25} - \)\(20\!\cdots\!86\)\( p^{110} T^{26} - \)\(52\!\cdots\!52\)\( p^{121} T^{27} + \)\(29\!\cdots\!27\)\( p^{132} T^{28} + \)\(22\!\cdots\!20\)\( p^{143} T^{29} - 253548316318958 p^{154} T^{30} - 6910556 p^{165} T^{31} + p^{176} T^{32} \)
23 \( 1 - 51387712 T - 956451396986360 T^{2} + \)\(64\!\cdots\!00\)\( T^{3} - \)\(30\!\cdots\!12\)\( T^{4} + \)\(34\!\cdots\!96\)\( T^{5} - \)\(69\!\cdots\!72\)\( T^{6} - \)\(39\!\cdots\!24\)\( T^{7} + \)\(88\!\cdots\!42\)\( T^{8} - \)\(16\!\cdots\!08\)\( T^{9} - \)\(55\!\cdots\!72\)\( T^{10} - \)\(50\!\cdots\!92\)\( T^{11} + \)\(58\!\cdots\!28\)\( p T^{12} + \)\(29\!\cdots\!80\)\( T^{13} - \)\(12\!\cdots\!20\)\( T^{14} - \)\(16\!\cdots\!80\)\( T^{15} - \)\(11\!\cdots\!97\)\( T^{16} - \)\(16\!\cdots\!80\)\( p^{11} T^{17} - \)\(12\!\cdots\!20\)\( p^{22} T^{18} + \)\(29\!\cdots\!80\)\( p^{33} T^{19} + \)\(58\!\cdots\!28\)\( p^{45} T^{20} - \)\(50\!\cdots\!92\)\( p^{55} T^{21} - \)\(55\!\cdots\!72\)\( p^{66} T^{22} - \)\(16\!\cdots\!08\)\( p^{77} T^{23} + \)\(88\!\cdots\!42\)\( p^{88} T^{24} - \)\(39\!\cdots\!24\)\( p^{99} T^{25} - \)\(69\!\cdots\!72\)\( p^{110} T^{26} + \)\(34\!\cdots\!96\)\( p^{121} T^{27} - \)\(30\!\cdots\!12\)\( p^{132} T^{28} + \)\(64\!\cdots\!00\)\( p^{143} T^{29} - 956451396986360 p^{154} T^{30} - 51387712 p^{165} T^{31} + p^{176} T^{32} \)
29 \( ( 1 + 59427308 T + 46226033876802234 T^{2} + \)\(28\!\cdots\!52\)\( T^{3} + \)\(11\!\cdots\!29\)\( T^{4} + \)\(70\!\cdots\!20\)\( T^{5} + \)\(21\!\cdots\!34\)\( T^{6} + \)\(11\!\cdots\!44\)\( T^{7} + \)\(29\!\cdots\!36\)\( T^{8} + \)\(11\!\cdots\!44\)\( p^{11} T^{9} + \)\(21\!\cdots\!34\)\( p^{22} T^{10} + \)\(70\!\cdots\!20\)\( p^{33} T^{11} + \)\(11\!\cdots\!29\)\( p^{44} T^{12} + \)\(28\!\cdots\!52\)\( p^{55} T^{13} + 46226033876802234 p^{66} T^{14} + 59427308 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
31 \( 1 - 164659160 T - 82319068117243904 T^{2} + \)\(22\!\cdots\!76\)\( T^{3} + \)\(14\!\cdots\!90\)\( T^{4} - \)\(10\!\cdots\!56\)\( T^{5} + \)\(64\!\cdots\!36\)\( T^{6} + \)\(11\!\cdots\!24\)\( T^{7} - \)\(11\!\cdots\!15\)\( T^{8} + \)\(34\!\cdots\!64\)\( T^{9} - \)\(13\!\cdots\!44\)\( T^{10} + \)\(24\!\cdots\!88\)\( T^{11} + \)\(58\!\cdots\!42\)\( T^{12} - \)\(75\!\cdots\!24\)\( T^{13} - \)\(45\!\cdots\!80\)\( T^{14} + \)\(13\!\cdots\!08\)\( T^{15} - \)\(11\!\cdots\!72\)\( T^{16} + \)\(13\!\cdots\!08\)\( p^{11} T^{17} - \)\(45\!\cdots\!80\)\( p^{22} T^{18} - \)\(75\!\cdots\!24\)\( p^{33} T^{19} + \)\(58\!\cdots\!42\)\( p^{44} T^{20} + \)\(24\!\cdots\!88\)\( p^{55} T^{21} - \)\(13\!\cdots\!44\)\( p^{66} T^{22} + \)\(34\!\cdots\!64\)\( p^{77} T^{23} - \)\(11\!\cdots\!15\)\( p^{88} T^{24} + \)\(11\!\cdots\!24\)\( p^{99} T^{25} + \)\(64\!\cdots\!36\)\( p^{110} T^{26} - \)\(10\!\cdots\!56\)\( p^{121} T^{27} + \)\(14\!\cdots\!90\)\( p^{132} T^{28} + \)\(22\!\cdots\!76\)\( p^{143} T^{29} - 82319068117243904 p^{154} T^{30} - 164659160 p^{165} T^{31} + p^{176} T^{32} \)
37 \( 1 - 75658364 T - 1105801172808372638 T^{2} + \)\(74\!\cdots\!52\)\( T^{3} + \)\(64\!\cdots\!03\)\( T^{4} - \)\(37\!\cdots\!60\)\( T^{5} - \)\(26\!\cdots\!10\)\( T^{6} + \)\(35\!\cdots\!36\)\( p T^{7} + \)\(90\!\cdots\!49\)\( T^{8} - \)\(34\!\cdots\!76\)\( T^{9} - \)\(25\!\cdots\!80\)\( T^{10} + \)\(69\!\cdots\!28\)\( T^{11} + \)\(62\!\cdots\!70\)\( T^{12} - \)\(99\!\cdots\!08\)\( T^{13} - \)\(13\!\cdots\!00\)\( T^{14} + \)\(68\!\cdots\!72\)\( T^{15} + \)\(25\!\cdots\!42\)\( T^{16} + \)\(68\!\cdots\!72\)\( p^{11} T^{17} - \)\(13\!\cdots\!00\)\( p^{22} T^{18} - \)\(99\!\cdots\!08\)\( p^{33} T^{19} + \)\(62\!\cdots\!70\)\( p^{44} T^{20} + \)\(69\!\cdots\!28\)\( p^{55} T^{21} - \)\(25\!\cdots\!80\)\( p^{66} T^{22} - \)\(34\!\cdots\!76\)\( p^{77} T^{23} + \)\(90\!\cdots\!49\)\( p^{88} T^{24} + \)\(35\!\cdots\!36\)\( p^{100} T^{25} - \)\(26\!\cdots\!10\)\( p^{110} T^{26} - \)\(37\!\cdots\!60\)\( p^{121} T^{27} + \)\(64\!\cdots\!03\)\( p^{132} T^{28} + \)\(74\!\cdots\!52\)\( p^{143} T^{29} - 1105801172808372638 p^{154} T^{30} - 75658364 p^{165} T^{31} + p^{176} T^{32} \)
41 \( ( 1 - 907784304 T + 2276369884381447456 T^{2} - \)\(88\!\cdots\!88\)\( T^{3} + \)\(17\!\cdots\!48\)\( T^{4} + \)\(64\!\cdots\!64\)\( T^{5} + \)\(80\!\cdots\!80\)\( T^{6} + \)\(34\!\cdots\!08\)\( T^{7} + \)\(40\!\cdots\!46\)\( T^{8} + \)\(34\!\cdots\!08\)\( p^{11} T^{9} + \)\(80\!\cdots\!80\)\( p^{22} T^{10} + \)\(64\!\cdots\!64\)\( p^{33} T^{11} + \)\(17\!\cdots\!48\)\( p^{44} T^{12} - \)\(88\!\cdots\!88\)\( p^{55} T^{13} + 2276369884381447456 p^{66} T^{14} - 907784304 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
43 \( ( 1 - 5377204 T + 3909451261563940470 T^{2} + \)\(52\!\cdots\!24\)\( T^{3} + \)\(80\!\cdots\!01\)\( T^{4} + \)\(18\!\cdots\!92\)\( T^{5} + \)\(11\!\cdots\!98\)\( T^{6} + \)\(29\!\cdots\!20\)\( T^{7} + \)\(12\!\cdots\!84\)\( T^{8} + \)\(29\!\cdots\!20\)\( p^{11} T^{9} + \)\(11\!\cdots\!98\)\( p^{22} T^{10} + \)\(18\!\cdots\!92\)\( p^{33} T^{11} + \)\(80\!\cdots\!01\)\( p^{44} T^{12} + \)\(52\!\cdots\!24\)\( p^{55} T^{13} + 3909451261563940470 p^{66} T^{14} - 5377204 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
47 \( 1 - 1034359464 T - 14866532978393700760 T^{2} + \)\(18\!\cdots\!24\)\( T^{3} + \)\(11\!\cdots\!60\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{5} - \)\(64\!\cdots\!64\)\( T^{6} + \)\(98\!\cdots\!32\)\( T^{7} + \)\(26\!\cdots\!22\)\( T^{8} - \)\(41\!\cdots\!20\)\( T^{9} - \)\(86\!\cdots\!28\)\( T^{10} + \)\(12\!\cdots\!40\)\( T^{11} + \)\(23\!\cdots\!00\)\( T^{12} - \)\(26\!\cdots\!68\)\( T^{13} - \)\(59\!\cdots\!20\)\( T^{14} + \)\(26\!\cdots\!48\)\( T^{15} + \)\(14\!\cdots\!83\)\( T^{16} + \)\(26\!\cdots\!48\)\( p^{11} T^{17} - \)\(59\!\cdots\!20\)\( p^{22} T^{18} - \)\(26\!\cdots\!68\)\( p^{33} T^{19} + \)\(23\!\cdots\!00\)\( p^{44} T^{20} + \)\(12\!\cdots\!40\)\( p^{55} T^{21} - \)\(86\!\cdots\!28\)\( p^{66} T^{22} - \)\(41\!\cdots\!20\)\( p^{77} T^{23} + \)\(26\!\cdots\!22\)\( p^{88} T^{24} + \)\(98\!\cdots\!32\)\( p^{99} T^{25} - \)\(64\!\cdots\!64\)\( p^{110} T^{26} - \)\(16\!\cdots\!68\)\( p^{121} T^{27} + \)\(11\!\cdots\!60\)\( p^{132} T^{28} + \)\(18\!\cdots\!24\)\( p^{143} T^{29} - 14866532978393700760 p^{154} T^{30} - 1034359464 p^{165} T^{31} + p^{176} T^{32} \)
53 \( 1 - 665159988 T - 34082090469894114658 T^{2} + \)\(15\!\cdots\!20\)\( T^{3} + \)\(51\!\cdots\!67\)\( T^{4} - \)\(12\!\cdots\!28\)\( T^{5} - \)\(47\!\cdots\!70\)\( T^{6} + \)\(11\!\cdots\!24\)\( T^{7} + \)\(32\!\cdots\!53\)\( T^{8} - \)\(81\!\cdots\!40\)\( T^{9} - \)\(21\!\cdots\!76\)\( T^{10} - \)\(17\!\cdots\!52\)\( T^{11} + \)\(15\!\cdots\!98\)\( T^{12} + \)\(43\!\cdots\!00\)\( T^{13} - \)\(10\!\cdots\!36\)\( T^{14} - \)\(23\!\cdots\!96\)\( T^{15} + \)\(85\!\cdots\!10\)\( T^{16} - \)\(23\!\cdots\!96\)\( p^{11} T^{17} - \)\(10\!\cdots\!36\)\( p^{22} T^{18} + \)\(43\!\cdots\!00\)\( p^{33} T^{19} + \)\(15\!\cdots\!98\)\( p^{44} T^{20} - \)\(17\!\cdots\!52\)\( p^{55} T^{21} - \)\(21\!\cdots\!76\)\( p^{66} T^{22} - \)\(81\!\cdots\!40\)\( p^{77} T^{23} + \)\(32\!\cdots\!53\)\( p^{88} T^{24} + \)\(11\!\cdots\!24\)\( p^{99} T^{25} - \)\(47\!\cdots\!70\)\( p^{110} T^{26} - \)\(12\!\cdots\!28\)\( p^{121} T^{27} + \)\(51\!\cdots\!67\)\( p^{132} T^{28} + \)\(15\!\cdots\!20\)\( p^{143} T^{29} - 34082090469894114658 p^{154} T^{30} - 665159988 p^{165} T^{31} + p^{176} T^{32} \)
59 \( 1 + 1040514580 T - \)\(14\!\cdots\!30\)\( T^{2} + \)\(52\!\cdots\!20\)\( T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} - \)\(60\!\cdots\!46\)\( T^{6} + \)\(17\!\cdots\!00\)\( T^{7} + \)\(21\!\cdots\!25\)\( T^{8} - \)\(10\!\cdots\!20\)\( T^{9} - \)\(48\!\cdots\!96\)\( T^{10} + \)\(41\!\cdots\!00\)\( T^{11} + \)\(36\!\cdots\!70\)\( T^{12} - \)\(11\!\cdots\!00\)\( T^{13} + \)\(22\!\cdots\!60\)\( T^{14} + \)\(13\!\cdots\!80\)\( T^{15} - \)\(10\!\cdots\!78\)\( T^{16} + \)\(13\!\cdots\!80\)\( p^{11} T^{17} + \)\(22\!\cdots\!60\)\( p^{22} T^{18} - \)\(11\!\cdots\!00\)\( p^{33} T^{19} + \)\(36\!\cdots\!70\)\( p^{44} T^{20} + \)\(41\!\cdots\!00\)\( p^{55} T^{21} - \)\(48\!\cdots\!96\)\( p^{66} T^{22} - \)\(10\!\cdots\!20\)\( p^{77} T^{23} + \)\(21\!\cdots\!25\)\( p^{88} T^{24} + \)\(17\!\cdots\!00\)\( p^{99} T^{25} - \)\(60\!\cdots\!46\)\( p^{110} T^{26} - \)\(17\!\cdots\!20\)\( p^{121} T^{27} + \)\(11\!\cdots\!83\)\( p^{132} T^{28} + \)\(52\!\cdots\!20\)\( p^{143} T^{29} - \)\(14\!\cdots\!30\)\( p^{154} T^{30} + 1040514580 p^{165} T^{31} + p^{176} T^{32} \)
61 \( 1 + 14391208024 T - \)\(11\!\cdots\!04\)\( T^{2} - \)\(23\!\cdots\!44\)\( T^{3} + \)\(89\!\cdots\!80\)\( T^{4} + \)\(24\!\cdots\!44\)\( T^{5} - \)\(36\!\cdots\!00\)\( T^{6} - \)\(19\!\cdots\!08\)\( T^{7} - \)\(80\!\cdots\!42\)\( T^{8} + \)\(11\!\cdots\!32\)\( T^{9} + \)\(26\!\cdots\!20\)\( T^{10} - \)\(51\!\cdots\!60\)\( T^{11} - \)\(26\!\cdots\!36\)\( T^{12} + \)\(17\!\cdots\!76\)\( T^{13} + \)\(16\!\cdots\!92\)\( T^{14} - \)\(28\!\cdots\!68\)\( T^{15} - \)\(83\!\cdots\!69\)\( T^{16} - \)\(28\!\cdots\!68\)\( p^{11} T^{17} + \)\(16\!\cdots\!92\)\( p^{22} T^{18} + \)\(17\!\cdots\!76\)\( p^{33} T^{19} - \)\(26\!\cdots\!36\)\( p^{44} T^{20} - \)\(51\!\cdots\!60\)\( p^{55} T^{21} + \)\(26\!\cdots\!20\)\( p^{66} T^{22} + \)\(11\!\cdots\!32\)\( p^{77} T^{23} - \)\(80\!\cdots\!42\)\( p^{88} T^{24} - \)\(19\!\cdots\!08\)\( p^{99} T^{25} - \)\(36\!\cdots\!00\)\( p^{110} T^{26} + \)\(24\!\cdots\!44\)\( p^{121} T^{27} + \)\(89\!\cdots\!80\)\( p^{132} T^{28} - \)\(23\!\cdots\!44\)\( p^{143} T^{29} - \)\(11\!\cdots\!04\)\( p^{154} T^{30} + 14391208024 p^{165} T^{31} + p^{176} T^{32} \)
67 \( 1 + 33307097284 T + 86676389803792405978 T^{2} - \)\(47\!\cdots\!64\)\( T^{3} + \)\(54\!\cdots\!55\)\( T^{4} + \)\(14\!\cdots\!28\)\( T^{5} - \)\(74\!\cdots\!54\)\( T^{6} - \)\(46\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!21\)\( T^{8} + \)\(29\!\cdots\!00\)\( T^{9} - \)\(21\!\cdots\!48\)\( T^{10} + \)\(51\!\cdots\!40\)\( T^{11} + \)\(30\!\cdots\!50\)\( T^{12} - \)\(36\!\cdots\!40\)\( T^{13} + \)\(54\!\cdots\!44\)\( T^{14} + \)\(55\!\cdots\!72\)\( T^{15} - \)\(43\!\cdots\!30\)\( T^{16} + \)\(55\!\cdots\!72\)\( p^{11} T^{17} + \)\(54\!\cdots\!44\)\( p^{22} T^{18} - \)\(36\!\cdots\!40\)\( p^{33} T^{19} + \)\(30\!\cdots\!50\)\( p^{44} T^{20} + \)\(51\!\cdots\!40\)\( p^{55} T^{21} - \)\(21\!\cdots\!48\)\( p^{66} T^{22} + \)\(29\!\cdots\!00\)\( p^{77} T^{23} + \)\(33\!\cdots\!21\)\( p^{88} T^{24} - \)\(46\!\cdots\!88\)\( p^{99} T^{25} - \)\(74\!\cdots\!54\)\( p^{110} T^{26} + \)\(14\!\cdots\!28\)\( p^{121} T^{27} + \)\(54\!\cdots\!55\)\( p^{132} T^{28} - \)\(47\!\cdots\!64\)\( p^{143} T^{29} + 86676389803792405978 p^{154} T^{30} + 33307097284 p^{165} T^{31} + p^{176} T^{32} \)
71 \( ( 1 + 32924451448 T + \)\(13\!\cdots\!56\)\( T^{2} + \)\(24\!\cdots\!00\)\( T^{3} + \)\(61\!\cdots\!96\)\( T^{4} + \)\(80\!\cdots\!56\)\( T^{5} + \)\(17\!\cdots\!08\)\( T^{6} + \)\(18\!\cdots\!48\)\( T^{7} + \)\(39\!\cdots\!10\)\( T^{8} + \)\(18\!\cdots\!48\)\( p^{11} T^{9} + \)\(17\!\cdots\!08\)\( p^{22} T^{10} + \)\(80\!\cdots\!56\)\( p^{33} T^{11} + \)\(61\!\cdots\!96\)\( p^{44} T^{12} + \)\(24\!\cdots\!00\)\( p^{55} T^{13} + \)\(13\!\cdots\!56\)\( p^{66} T^{14} + 32924451448 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
73 \( 1 - 17709749204 T - \)\(96\!\cdots\!42\)\( T^{2} + \)\(79\!\cdots\!40\)\( T^{3} + \)\(58\!\cdots\!03\)\( T^{4} + \)\(21\!\cdots\!88\)\( T^{5} - \)\(24\!\cdots\!14\)\( T^{6} - \)\(35\!\cdots\!28\)\( T^{7} + \)\(72\!\cdots\!25\)\( T^{8} + \)\(19\!\cdots\!12\)\( T^{9} - \)\(10\!\cdots\!32\)\( T^{10} - \)\(73\!\cdots\!24\)\( T^{11} - \)\(38\!\cdots\!86\)\( T^{12} + \)\(19\!\cdots\!80\)\( T^{13} + \)\(34\!\cdots\!56\)\( T^{14} - \)\(24\!\cdots\!76\)\( T^{15} - \)\(13\!\cdots\!22\)\( T^{16} - \)\(24\!\cdots\!76\)\( p^{11} T^{17} + \)\(34\!\cdots\!56\)\( p^{22} T^{18} + \)\(19\!\cdots\!80\)\( p^{33} T^{19} - \)\(38\!\cdots\!86\)\( p^{44} T^{20} - \)\(73\!\cdots\!24\)\( p^{55} T^{21} - \)\(10\!\cdots\!32\)\( p^{66} T^{22} + \)\(19\!\cdots\!12\)\( p^{77} T^{23} + \)\(72\!\cdots\!25\)\( p^{88} T^{24} - \)\(35\!\cdots\!28\)\( p^{99} T^{25} - \)\(24\!\cdots\!14\)\( p^{110} T^{26} + \)\(21\!\cdots\!88\)\( p^{121} T^{27} + \)\(58\!\cdots\!03\)\( p^{132} T^{28} + \)\(79\!\cdots\!40\)\( p^{143} T^{29} - \)\(96\!\cdots\!42\)\( p^{154} T^{30} - 17709749204 p^{165} T^{31} + p^{176} T^{32} \)
79 \( 1 + 26626784032 T - \)\(28\!\cdots\!56\)\( T^{2} - \)\(83\!\cdots\!60\)\( T^{3} + \)\(37\!\cdots\!30\)\( T^{4} + \)\(11\!\cdots\!96\)\( T^{5} - \)\(35\!\cdots\!04\)\( T^{6} - \)\(10\!\cdots\!76\)\( T^{7} + \)\(33\!\cdots\!61\)\( T^{8} + \)\(78\!\cdots\!28\)\( T^{9} - \)\(30\!\cdots\!96\)\( T^{10} - \)\(55\!\cdots\!08\)\( T^{11} + \)\(22\!\cdots\!42\)\( T^{12} + \)\(31\!\cdots\!28\)\( T^{13} - \)\(15\!\cdots\!60\)\( T^{14} - \)\(88\!\cdots\!16\)\( T^{15} + \)\(10\!\cdots\!20\)\( T^{16} - \)\(88\!\cdots\!16\)\( p^{11} T^{17} - \)\(15\!\cdots\!60\)\( p^{22} T^{18} + \)\(31\!\cdots\!28\)\( p^{33} T^{19} + \)\(22\!\cdots\!42\)\( p^{44} T^{20} - \)\(55\!\cdots\!08\)\( p^{55} T^{21} - \)\(30\!\cdots\!96\)\( p^{66} T^{22} + \)\(78\!\cdots\!28\)\( p^{77} T^{23} + \)\(33\!\cdots\!61\)\( p^{88} T^{24} - \)\(10\!\cdots\!76\)\( p^{99} T^{25} - \)\(35\!\cdots\!04\)\( p^{110} T^{26} + \)\(11\!\cdots\!96\)\( p^{121} T^{27} + \)\(37\!\cdots\!30\)\( p^{132} T^{28} - \)\(83\!\cdots\!60\)\( p^{143} T^{29} - \)\(28\!\cdots\!56\)\( p^{154} T^{30} + 26626784032 p^{165} T^{31} + p^{176} T^{32} \)
83 \( ( 1 - 105153477524 T + \)\(97\!\cdots\!22\)\( T^{2} - \)\(61\!\cdots\!20\)\( T^{3} + \)\(37\!\cdots\!01\)\( T^{4} - \)\(17\!\cdots\!48\)\( T^{5} + \)\(81\!\cdots\!38\)\( T^{6} - \)\(32\!\cdots\!00\)\( T^{7} + \)\(12\!\cdots\!80\)\( T^{8} - \)\(32\!\cdots\!00\)\( p^{11} T^{9} + \)\(81\!\cdots\!38\)\( p^{22} T^{10} - \)\(17\!\cdots\!48\)\( p^{33} T^{11} + \)\(37\!\cdots\!01\)\( p^{44} T^{12} - \)\(61\!\cdots\!20\)\( p^{55} T^{13} + \)\(97\!\cdots\!22\)\( p^{66} T^{14} - 105153477524 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
89 \( 1 - 55951560072 T - \)\(10\!\cdots\!44\)\( T^{2} + \)\(37\!\cdots\!68\)\( T^{3} + \)\(78\!\cdots\!60\)\( T^{4} - \)\(16\!\cdots\!12\)\( T^{5} - \)\(35\!\cdots\!20\)\( T^{6} + \)\(24\!\cdots\!96\)\( T^{7} + \)\(10\!\cdots\!66\)\( T^{8} + \)\(62\!\cdots\!40\)\( T^{9} - \)\(14\!\cdots\!80\)\( T^{10} - \)\(57\!\cdots\!32\)\( T^{11} - \)\(40\!\cdots\!04\)\( T^{12} + \)\(16\!\cdots\!36\)\( T^{13} + \)\(34\!\cdots\!40\)\( T^{14} - \)\(21\!\cdots\!40\)\( T^{15} - \)\(12\!\cdots\!05\)\( T^{16} - \)\(21\!\cdots\!40\)\( p^{11} T^{17} + \)\(34\!\cdots\!40\)\( p^{22} T^{18} + \)\(16\!\cdots\!36\)\( p^{33} T^{19} - \)\(40\!\cdots\!04\)\( p^{44} T^{20} - \)\(57\!\cdots\!32\)\( p^{55} T^{21} - \)\(14\!\cdots\!80\)\( p^{66} T^{22} + \)\(62\!\cdots\!40\)\( p^{77} T^{23} + \)\(10\!\cdots\!66\)\( p^{88} T^{24} + \)\(24\!\cdots\!96\)\( p^{99} T^{25} - \)\(35\!\cdots\!20\)\( p^{110} T^{26} - \)\(16\!\cdots\!12\)\( p^{121} T^{27} + \)\(78\!\cdots\!60\)\( p^{132} T^{28} + \)\(37\!\cdots\!68\)\( p^{143} T^{29} - \)\(10\!\cdots\!44\)\( p^{154} T^{30} - 55951560072 p^{165} T^{31} + p^{176} T^{32} \)
97 \( ( 1 + 78108015356 T + \)\(15\!\cdots\!70\)\( T^{2} + \)\(93\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!81\)\( T^{4} + \)\(59\!\cdots\!44\)\( T^{5} + \)\(73\!\cdots\!30\)\( T^{6} + \)\(86\!\cdots\!68\)\( T^{7} + \)\(53\!\cdots\!92\)\( T^{8} + \)\(86\!\cdots\!68\)\( p^{11} T^{9} + \)\(73\!\cdots\!30\)\( p^{22} T^{10} + \)\(59\!\cdots\!44\)\( p^{33} T^{11} + \)\(16\!\cdots\!81\)\( p^{44} T^{12} + \)\(93\!\cdots\!56\)\( p^{55} T^{13} + \)\(15\!\cdots\!70\)\( p^{66} T^{14} + 78108015356 p^{77} T^{15} + p^{88} T^{16} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−1.76010811611861681082616926964, −1.68973818242346126199257361248, −1.66702598156921249451612066197, −1.66191399229931823655731984680, −1.51361466176660098069319843229, −1.49970955427997187958011239820, −1.40471053283801573920350541802, −1.33580622600207863936060827527, −1.11322957608700849650511990241, −1.07517874771252213993681255336, −1.01706927005932234262741629198, −1.00644817602059270456660210377, −0.929671569547179210966137904742, −0.922647806466379570406600765826, −0.917944685101398425773061501388, −0.67737367748037752494171566098, −0.66579353185863972853120481445, −0.65263963380296872451600853685, −0.54572271643007910276577895571, −0.53195894322323973344126076514, −0.38892313228446263877155911542, −0.28188280329616078773459696754, −0.13618873718443771924240044316, −0.093478255536231775608334972651, −0.088875144554082774384085114930, 0.088875144554082774384085114930, 0.093478255536231775608334972651, 0.13618873718443771924240044316, 0.28188280329616078773459696754, 0.38892313228446263877155911542, 0.53195894322323973344126076514, 0.54572271643007910276577895571, 0.65263963380296872451600853685, 0.66579353185863972853120481445, 0.67737367748037752494171566098, 0.917944685101398425773061501388, 0.922647806466379570406600765826, 0.929671569547179210966137904742, 1.00644817602059270456660210377, 1.01706927005932234262741629198, 1.07517874771252213993681255336, 1.11322957608700849650511990241, 1.33580622600207863936060827527, 1.40471053283801573920350541802, 1.49970955427997187958011239820, 1.51361466176660098069319843229, 1.66191399229931823655731984680, 1.66702598156921249451612066197, 1.68973818242346126199257361248, 1.76010811611861681082616926964

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

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