Properties

Label 32-273e16-1.1-c11e16-0-0
Degree $32$
Conductor $9.519\times 10^{38}$
Sign $1$
Analytic cond. $1.40438\times 10^{37}$
Root an. cond. $14.4830$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $16$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 63·2-s − 3.88e3·3-s − 4.01e3·4-s − 3.16e3·5-s + 2.44e5·6-s + 2.68e5·7-s + 3.16e5·8-s + 8.03e6·9-s + 1.99e5·10-s − 4.66e5·11-s + 1.56e7·12-s − 5.94e6·13-s − 1.69e7·14-s + 1.23e7·15-s + 7.38e6·16-s + 1.75e6·17-s − 5.05e8·18-s + 4.23e6·19-s + 1.27e7·20-s − 1.04e9·21-s + 2.94e7·22-s − 1.50e8·23-s − 1.23e9·24-s − 3.10e8·25-s + 3.74e8·26-s − 1.17e10·27-s − 1.08e9·28-s + ⋯
L(s)  = 1  − 1.39·2-s − 9.23·3-s − 1.96·4-s − 0.453·5-s + 12.8·6-s + 6.04·7-s + 3.41·8-s + 45.3·9-s + 0.631·10-s − 0.874·11-s + 18.1·12-s − 4.43·13-s − 8.41·14-s + 4.18·15-s + 1.76·16-s + 0.299·17-s − 63.1·18-s + 0.392·19-s + 0.889·20-s − 55.8·21-s + 1.21·22-s − 4.87·23-s − 31.5·24-s − 6.35·25-s + 6.17·26-s − 157.·27-s − 11.8·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(12-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+11/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(32\)
Conductor: \(3^{16} \cdot 7^{16} \cdot 13^{16}\)
Sign: $1$
Analytic conductor: \(1.40438\times 10^{37}\)
Root analytic conductor: \(14.4830\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{273} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(16\)
Selberg data: \((32,\ 3^{16} \cdot 7^{16} \cdot 13^{16} ,\ ( \ : [11/2]^{16} ),\ 1 )\)

Particular Values

\(L(6)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{5} T )^{16} \)
7 \( ( 1 - p^{5} T )^{16} \)
13 \( ( 1 + p^{5} T )^{16} \)
good2 \( 1 + 63 T + 1997 p^{2} T^{2} + 219771 p T^{3} + 4055519 p^{3} T^{4} + 6384663 p^{8} T^{5} + 3037487649 p^{5} T^{6} + 144569617173 p^{5} T^{7} + 482873893431 p^{9} T^{8} + 41210051108841 p^{8} T^{9} + 60521892083273 p^{13} T^{10} + 2239031303654031 p^{13} T^{11} + 11791744845855377 p^{16} T^{12} + 23806191912815115 p^{20} T^{13} + 68891401551621091 p^{24} T^{14} + 1023283383416203185 p^{25} T^{15} + 7892242921598689875 p^{28} T^{16} + 1023283383416203185 p^{36} T^{17} + 68891401551621091 p^{46} T^{18} + 23806191912815115 p^{53} T^{19} + 11791744845855377 p^{60} T^{20} + 2239031303654031 p^{68} T^{21} + 60521892083273 p^{79} T^{22} + 41210051108841 p^{85} T^{23} + 482873893431 p^{97} T^{24} + 144569617173 p^{104} T^{25} + 3037487649 p^{115} T^{26} + 6384663 p^{129} T^{27} + 4055519 p^{135} T^{28} + 219771 p^{144} T^{29} + 1997 p^{156} T^{30} + 63 p^{165} T^{31} + p^{176} T^{32} \)
5 \( 1 + 3168 T + 320151524 T^{2} + 1023097332312 T^{3} + 52139051805256326 T^{4} + 32871579807780458208 p T^{5} + \)\(22\!\cdots\!56\)\( p^{2} T^{6} + \)\(14\!\cdots\!92\)\( p^{3} T^{7} + \)\(75\!\cdots\!21\)\( p^{4} T^{8} + \)\(99\!\cdots\!64\)\( p^{6} T^{9} + \)\(79\!\cdots\!68\)\( p^{8} T^{10} + \)\(14\!\cdots\!32\)\( p^{7} T^{11} + \)\(44\!\cdots\!18\)\( p^{8} T^{12} + \)\(35\!\cdots\!68\)\( p^{9} T^{13} + \)\(90\!\cdots\!04\)\( p^{10} T^{14} + \)\(15\!\cdots\!84\)\( p^{12} T^{15} + \)\(70\!\cdots\!24\)\( p^{14} T^{16} + \)\(15\!\cdots\!84\)\( p^{23} T^{17} + \)\(90\!\cdots\!04\)\( p^{32} T^{18} + \)\(35\!\cdots\!68\)\( p^{42} T^{19} + \)\(44\!\cdots\!18\)\( p^{52} T^{20} + \)\(14\!\cdots\!32\)\( p^{62} T^{21} + \)\(79\!\cdots\!68\)\( p^{74} T^{22} + \)\(99\!\cdots\!64\)\( p^{83} T^{23} + \)\(75\!\cdots\!21\)\( p^{92} T^{24} + \)\(14\!\cdots\!92\)\( p^{102} T^{25} + \)\(22\!\cdots\!56\)\( p^{112} T^{26} + 32871579807780458208 p^{122} T^{27} + 52139051805256326 p^{132} T^{28} + 1023097332312 p^{143} T^{29} + 320151524 p^{154} T^{30} + 3168 p^{165} T^{31} + p^{176} T^{32} \)
11 \( 1 + 42444 p T + 1819831942074 T^{2} + 77178495131611128 p T^{3} + \)\(16\!\cdots\!05\)\( T^{4} + \)\(67\!\cdots\!24\)\( p T^{5} + \)\(10\!\cdots\!70\)\( T^{6} + \)\(38\!\cdots\!84\)\( p T^{7} + \)\(52\!\cdots\!10\)\( T^{8} + \)\(16\!\cdots\!32\)\( p T^{9} + \)\(21\!\cdots\!58\)\( T^{10} + \)\(64\!\cdots\!12\)\( p T^{11} + \)\(80\!\cdots\!23\)\( T^{12} + \)\(22\!\cdots\!12\)\( p T^{13} + \)\(26\!\cdots\!14\)\( T^{14} + \)\(70\!\cdots\!52\)\( p T^{15} + \)\(80\!\cdots\!94\)\( T^{16} + \)\(70\!\cdots\!52\)\( p^{12} T^{17} + \)\(26\!\cdots\!14\)\( p^{22} T^{18} + \)\(22\!\cdots\!12\)\( p^{34} T^{19} + \)\(80\!\cdots\!23\)\( p^{44} T^{20} + \)\(64\!\cdots\!12\)\( p^{56} T^{21} + \)\(21\!\cdots\!58\)\( p^{66} T^{22} + \)\(16\!\cdots\!32\)\( p^{78} T^{23} + \)\(52\!\cdots\!10\)\( p^{88} T^{24} + \)\(38\!\cdots\!84\)\( p^{100} T^{25} + \)\(10\!\cdots\!70\)\( p^{110} T^{26} + \)\(67\!\cdots\!24\)\( p^{122} T^{27} + \)\(16\!\cdots\!05\)\( p^{132} T^{28} + 77178495131611128 p^{144} T^{29} + 1819831942074 p^{154} T^{30} + 42444 p^{166} T^{31} + p^{176} T^{32} \)
17 \( 1 - 1753452 T + 272778464406782 T^{2} - \)\(13\!\cdots\!20\)\( T^{3} + \)\(22\!\cdots\!53\)\( p T^{4} + \)\(25\!\cdots\!00\)\( T^{5} + \)\(20\!\cdots\!38\)\( p T^{6} + \)\(64\!\cdots\!80\)\( T^{7} + \)\(24\!\cdots\!78\)\( T^{8} + \)\(72\!\cdots\!00\)\( T^{9} + \)\(13\!\cdots\!58\)\( T^{10} + \)\(54\!\cdots\!16\)\( T^{11} + \)\(36\!\cdots\!55\)\( p T^{12} + \)\(30\!\cdots\!20\)\( T^{13} + \)\(87\!\cdots\!58\)\( p^{2} T^{14} + \)\(13\!\cdots\!80\)\( T^{15} + \)\(91\!\cdots\!42\)\( T^{16} + \)\(13\!\cdots\!80\)\( p^{11} T^{17} + \)\(87\!\cdots\!58\)\( p^{24} T^{18} + \)\(30\!\cdots\!20\)\( p^{33} T^{19} + \)\(36\!\cdots\!55\)\( p^{45} T^{20} + \)\(54\!\cdots\!16\)\( p^{55} T^{21} + \)\(13\!\cdots\!58\)\( p^{66} T^{22} + \)\(72\!\cdots\!00\)\( p^{77} T^{23} + \)\(24\!\cdots\!78\)\( p^{88} T^{24} + \)\(64\!\cdots\!80\)\( p^{99} T^{25} + \)\(20\!\cdots\!38\)\( p^{111} T^{26} + \)\(25\!\cdots\!00\)\( p^{121} T^{27} + \)\(22\!\cdots\!53\)\( p^{133} T^{28} - \)\(13\!\cdots\!20\)\( p^{143} T^{29} + 272778464406782 p^{154} T^{30} - 1753452 p^{165} T^{31} + p^{176} T^{32} \)
19 \( 1 - 4237800 T + 1119294184111812 T^{2} - \)\(59\!\cdots\!80\)\( T^{3} + \)\(63\!\cdots\!94\)\( T^{4} - \)\(38\!\cdots\!04\)\( T^{5} + \)\(24\!\cdots\!36\)\( T^{6} - \)\(15\!\cdots\!28\)\( T^{7} + \)\(69\!\cdots\!69\)\( T^{8} - \)\(46\!\cdots\!92\)\( T^{9} + \)\(15\!\cdots\!00\)\( T^{10} - \)\(10\!\cdots\!48\)\( T^{11} + \)\(28\!\cdots\!86\)\( T^{12} - \)\(18\!\cdots\!12\)\( T^{13} + \)\(43\!\cdots\!92\)\( T^{14} - \)\(26\!\cdots\!80\)\( T^{15} + \)\(55\!\cdots\!28\)\( T^{16} - \)\(26\!\cdots\!80\)\( p^{11} T^{17} + \)\(43\!\cdots\!92\)\( p^{22} T^{18} - \)\(18\!\cdots\!12\)\( p^{33} T^{19} + \)\(28\!\cdots\!86\)\( p^{44} T^{20} - \)\(10\!\cdots\!48\)\( p^{55} T^{21} + \)\(15\!\cdots\!00\)\( p^{66} T^{22} - \)\(46\!\cdots\!92\)\( p^{77} T^{23} + \)\(69\!\cdots\!69\)\( p^{88} T^{24} - \)\(15\!\cdots\!28\)\( p^{99} T^{25} + \)\(24\!\cdots\!36\)\( p^{110} T^{26} - \)\(38\!\cdots\!04\)\( p^{121} T^{27} + \)\(63\!\cdots\!94\)\( p^{132} T^{28} - \)\(59\!\cdots\!80\)\( p^{143} T^{29} + 1119294184111812 p^{154} T^{30} - 4237800 p^{165} T^{31} + p^{176} T^{32} \)
23 \( 1 + 150481440 T + 18318396308951624 T^{2} + \)\(15\!\cdots\!96\)\( T^{3} + \)\(10\!\cdots\!34\)\( T^{4} + \)\(64\!\cdots\!96\)\( T^{5} + \)\(34\!\cdots\!84\)\( T^{6} + \)\(16\!\cdots\!00\)\( T^{7} + \)\(70\!\cdots\!17\)\( T^{8} + \)\(27\!\cdots\!76\)\( T^{9} + \)\(10\!\cdots\!84\)\( T^{10} + \)\(33\!\cdots\!28\)\( T^{11} + \)\(11\!\cdots\!58\)\( T^{12} + \)\(33\!\cdots\!16\)\( T^{13} + \)\(10\!\cdots\!52\)\( T^{14} + \)\(30\!\cdots\!72\)\( T^{15} + \)\(93\!\cdots\!00\)\( T^{16} + \)\(30\!\cdots\!72\)\( p^{11} T^{17} + \)\(10\!\cdots\!52\)\( p^{22} T^{18} + \)\(33\!\cdots\!16\)\( p^{33} T^{19} + \)\(11\!\cdots\!58\)\( p^{44} T^{20} + \)\(33\!\cdots\!28\)\( p^{55} T^{21} + \)\(10\!\cdots\!84\)\( p^{66} T^{22} + \)\(27\!\cdots\!76\)\( p^{77} T^{23} + \)\(70\!\cdots\!17\)\( p^{88} T^{24} + \)\(16\!\cdots\!00\)\( p^{99} T^{25} + \)\(34\!\cdots\!84\)\( p^{110} T^{26} + \)\(64\!\cdots\!96\)\( p^{121} T^{27} + \)\(10\!\cdots\!34\)\( p^{132} T^{28} + \)\(15\!\cdots\!96\)\( p^{143} T^{29} + 18318396308951624 p^{154} T^{30} + 150481440 p^{165} T^{31} + p^{176} T^{32} \)
29 \( 1 + 111411432 T + 140907589341449384 T^{2} + \)\(12\!\cdots\!04\)\( T^{3} + \)\(93\!\cdots\!98\)\( T^{4} + \)\(65\!\cdots\!08\)\( T^{5} + \)\(39\!\cdots\!28\)\( T^{6} + \)\(20\!\cdots\!36\)\( T^{7} + \)\(11\!\cdots\!25\)\( T^{8} + \)\(42\!\cdots\!12\)\( T^{9} + \)\(26\!\cdots\!56\)\( T^{10} + \)\(60\!\cdots\!92\)\( T^{11} + \)\(48\!\cdots\!70\)\( T^{12} + \)\(57\!\cdots\!04\)\( T^{13} + \)\(74\!\cdots\!64\)\( T^{14} + \)\(44\!\cdots\!24\)\( T^{15} + \)\(97\!\cdots\!96\)\( T^{16} + \)\(44\!\cdots\!24\)\( p^{11} T^{17} + \)\(74\!\cdots\!64\)\( p^{22} T^{18} + \)\(57\!\cdots\!04\)\( p^{33} T^{19} + \)\(48\!\cdots\!70\)\( p^{44} T^{20} + \)\(60\!\cdots\!92\)\( p^{55} T^{21} + \)\(26\!\cdots\!56\)\( p^{66} T^{22} + \)\(42\!\cdots\!12\)\( p^{77} T^{23} + \)\(11\!\cdots\!25\)\( p^{88} T^{24} + \)\(20\!\cdots\!36\)\( p^{99} T^{25} + \)\(39\!\cdots\!28\)\( p^{110} T^{26} + \)\(65\!\cdots\!08\)\( p^{121} T^{27} + \)\(93\!\cdots\!98\)\( p^{132} T^{28} + \)\(12\!\cdots\!04\)\( p^{143} T^{29} + 140907589341449384 p^{154} T^{30} + 111411432 p^{165} T^{31} + p^{176} T^{32} \)
31 \( 1 + 90536236 T + 170987389409218902 T^{2} + \)\(94\!\cdots\!36\)\( T^{3} + \)\(14\!\cdots\!17\)\( T^{4} + \)\(42\!\cdots\!12\)\( T^{5} + \)\(78\!\cdots\!54\)\( T^{6} + \)\(85\!\cdots\!88\)\( T^{7} + \)\(33\!\cdots\!66\)\( T^{8} + \)\(12\!\cdots\!00\)\( T^{9} + \)\(11\!\cdots\!38\)\( T^{10} - \)\(28\!\cdots\!44\)\( T^{11} + \)\(37\!\cdots\!03\)\( T^{12} + \)\(22\!\cdots\!88\)\( T^{13} + \)\(10\!\cdots\!26\)\( T^{14} + \)\(13\!\cdots\!24\)\( T^{15} + \)\(28\!\cdots\!26\)\( T^{16} + \)\(13\!\cdots\!24\)\( p^{11} T^{17} + \)\(10\!\cdots\!26\)\( p^{22} T^{18} + \)\(22\!\cdots\!88\)\( p^{33} T^{19} + \)\(37\!\cdots\!03\)\( p^{44} T^{20} - \)\(28\!\cdots\!44\)\( p^{55} T^{21} + \)\(11\!\cdots\!38\)\( p^{66} T^{22} + \)\(12\!\cdots\!00\)\( p^{77} T^{23} + \)\(33\!\cdots\!66\)\( p^{88} T^{24} + \)\(85\!\cdots\!88\)\( p^{99} T^{25} + \)\(78\!\cdots\!54\)\( p^{110} T^{26} + \)\(42\!\cdots\!12\)\( p^{121} T^{27} + \)\(14\!\cdots\!17\)\( p^{132} T^{28} + \)\(94\!\cdots\!36\)\( p^{143} T^{29} + 170987389409218902 p^{154} T^{30} + 90536236 p^{165} T^{31} + p^{176} T^{32} \)
37 \( 1 + 1756337900 T + 3428636590713366786 T^{2} + \)\(41\!\cdots\!56\)\( T^{3} + \)\(49\!\cdots\!53\)\( T^{4} + \)\(47\!\cdots\!92\)\( T^{5} + \)\(42\!\cdots\!70\)\( T^{6} + \)\(33\!\cdots\!36\)\( T^{7} + \)\(25\!\cdots\!90\)\( T^{8} + \)\(16\!\cdots\!04\)\( T^{9} + \)\(10\!\cdots\!10\)\( T^{10} + \)\(63\!\cdots\!92\)\( T^{11} + \)\(35\!\cdots\!95\)\( T^{12} + \)\(18\!\cdots\!44\)\( T^{13} + \)\(88\!\cdots\!46\)\( T^{14} + \)\(11\!\cdots\!88\)\( p T^{15} + \)\(17\!\cdots\!82\)\( T^{16} + \)\(11\!\cdots\!88\)\( p^{12} T^{17} + \)\(88\!\cdots\!46\)\( p^{22} T^{18} + \)\(18\!\cdots\!44\)\( p^{33} T^{19} + \)\(35\!\cdots\!95\)\( p^{44} T^{20} + \)\(63\!\cdots\!92\)\( p^{55} T^{21} + \)\(10\!\cdots\!10\)\( p^{66} T^{22} + \)\(16\!\cdots\!04\)\( p^{77} T^{23} + \)\(25\!\cdots\!90\)\( p^{88} T^{24} + \)\(33\!\cdots\!36\)\( p^{99} T^{25} + \)\(42\!\cdots\!70\)\( p^{110} T^{26} + \)\(47\!\cdots\!92\)\( p^{121} T^{27} + \)\(49\!\cdots\!53\)\( p^{132} T^{28} + \)\(41\!\cdots\!56\)\( p^{143} T^{29} + 3428636590713366786 p^{154} T^{30} + 1756337900 p^{165} T^{31} + p^{176} T^{32} \)
41 \( 1 - 649575720 T + 5412000906547565728 T^{2} - \)\(36\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!96\)\( T^{4} - \)\(10\!\cdots\!36\)\( T^{5} + \)\(26\!\cdots\!24\)\( T^{6} - \)\(18\!\cdots\!32\)\( T^{7} + \)\(35\!\cdots\!52\)\( T^{8} - \)\(23\!\cdots\!40\)\( T^{9} + \)\(37\!\cdots\!56\)\( T^{10} - \)\(23\!\cdots\!44\)\( T^{11} + \)\(32\!\cdots\!84\)\( T^{12} - \)\(19\!\cdots\!20\)\( T^{13} + \)\(22\!\cdots\!52\)\( T^{14} - \)\(12\!\cdots\!28\)\( T^{15} + \)\(13\!\cdots\!34\)\( T^{16} - \)\(12\!\cdots\!28\)\( p^{11} T^{17} + \)\(22\!\cdots\!52\)\( p^{22} T^{18} - \)\(19\!\cdots\!20\)\( p^{33} T^{19} + \)\(32\!\cdots\!84\)\( p^{44} T^{20} - \)\(23\!\cdots\!44\)\( p^{55} T^{21} + \)\(37\!\cdots\!56\)\( p^{66} T^{22} - \)\(23\!\cdots\!40\)\( p^{77} T^{23} + \)\(35\!\cdots\!52\)\( p^{88} T^{24} - \)\(18\!\cdots\!32\)\( p^{99} T^{25} + \)\(26\!\cdots\!24\)\( p^{110} T^{26} - \)\(10\!\cdots\!36\)\( p^{121} T^{27} + \)\(14\!\cdots\!96\)\( p^{132} T^{28} - \)\(36\!\cdots\!40\)\( p^{143} T^{29} + 5412000906547565728 p^{154} T^{30} - 649575720 p^{165} T^{31} + p^{176} T^{32} \)
43 \( 1 + 1889554520 T + 10463376342718153752 T^{2} + \)\(16\!\cdots\!68\)\( T^{3} + \)\(51\!\cdots\!62\)\( T^{4} + \)\(72\!\cdots\!92\)\( T^{5} + \)\(15\!\cdots\!08\)\( T^{6} + \)\(19\!\cdots\!80\)\( T^{7} + \)\(35\!\cdots\!41\)\( T^{8} + \)\(39\!\cdots\!36\)\( T^{9} + \)\(59\!\cdots\!92\)\( T^{10} + \)\(61\!\cdots\!56\)\( T^{11} + \)\(81\!\cdots\!66\)\( T^{12} + \)\(77\!\cdots\!52\)\( T^{13} + \)\(94\!\cdots\!20\)\( T^{14} + \)\(83\!\cdots\!16\)\( T^{15} + \)\(21\!\cdots\!60\)\( p T^{16} + \)\(83\!\cdots\!16\)\( p^{11} T^{17} + \)\(94\!\cdots\!20\)\( p^{22} T^{18} + \)\(77\!\cdots\!52\)\( p^{33} T^{19} + \)\(81\!\cdots\!66\)\( p^{44} T^{20} + \)\(61\!\cdots\!56\)\( p^{55} T^{21} + \)\(59\!\cdots\!92\)\( p^{66} T^{22} + \)\(39\!\cdots\!36\)\( p^{77} T^{23} + \)\(35\!\cdots\!41\)\( p^{88} T^{24} + \)\(19\!\cdots\!80\)\( p^{99} T^{25} + \)\(15\!\cdots\!08\)\( p^{110} T^{26} + \)\(72\!\cdots\!92\)\( p^{121} T^{27} + \)\(51\!\cdots\!62\)\( p^{132} T^{28} + \)\(16\!\cdots\!68\)\( p^{143} T^{29} + 10463376342718153752 p^{154} T^{30} + 1889554520 p^{165} T^{31} + p^{176} T^{32} \)
47 \( 1 + 4036082940 T + 27322802044134664402 T^{2} + \)\(77\!\cdots\!88\)\( T^{3} + \)\(30\!\cdots\!25\)\( T^{4} + \)\(67\!\cdots\!36\)\( T^{5} + \)\(20\!\cdots\!46\)\( T^{6} + \)\(34\!\cdots\!52\)\( T^{7} + \)\(88\!\cdots\!30\)\( T^{8} + \)\(11\!\cdots\!52\)\( T^{9} + \)\(27\!\cdots\!10\)\( T^{10} + \)\(24\!\cdots\!76\)\( T^{11} + \)\(61\!\cdots\!99\)\( T^{12} + \)\(26\!\cdots\!12\)\( T^{13} + \)\(11\!\cdots\!34\)\( T^{14} - \)\(10\!\cdots\!40\)\( T^{15} + \)\(23\!\cdots\!34\)\( T^{16} - \)\(10\!\cdots\!40\)\( p^{11} T^{17} + \)\(11\!\cdots\!34\)\( p^{22} T^{18} + \)\(26\!\cdots\!12\)\( p^{33} T^{19} + \)\(61\!\cdots\!99\)\( p^{44} T^{20} + \)\(24\!\cdots\!76\)\( p^{55} T^{21} + \)\(27\!\cdots\!10\)\( p^{66} T^{22} + \)\(11\!\cdots\!52\)\( p^{77} T^{23} + \)\(88\!\cdots\!30\)\( p^{88} T^{24} + \)\(34\!\cdots\!52\)\( p^{99} T^{25} + \)\(20\!\cdots\!46\)\( p^{110} T^{26} + \)\(67\!\cdots\!36\)\( p^{121} T^{27} + \)\(30\!\cdots\!25\)\( p^{132} T^{28} + \)\(77\!\cdots\!88\)\( p^{143} T^{29} + 27322802044134664402 p^{154} T^{30} + 4036082940 p^{165} T^{31} + p^{176} T^{32} \)
53 \( 1 - 511144020 T + 75439096087978483550 T^{2} - \)\(43\!\cdots\!56\)\( T^{3} + \)\(28\!\cdots\!13\)\( T^{4} + \)\(98\!\cdots\!24\)\( T^{5} + \)\(72\!\cdots\!58\)\( T^{6} + \)\(52\!\cdots\!36\)\( T^{7} + \)\(13\!\cdots\!78\)\( T^{8} + \)\(14\!\cdots\!84\)\( T^{9} + \)\(21\!\cdots\!46\)\( T^{10} + \)\(29\!\cdots\!96\)\( T^{11} + \)\(27\!\cdots\!47\)\( T^{12} + \)\(43\!\cdots\!40\)\( T^{13} + \)\(30\!\cdots\!22\)\( T^{14} + \)\(50\!\cdots\!80\)\( T^{15} + \)\(29\!\cdots\!02\)\( T^{16} + \)\(50\!\cdots\!80\)\( p^{11} T^{17} + \)\(30\!\cdots\!22\)\( p^{22} T^{18} + \)\(43\!\cdots\!40\)\( p^{33} T^{19} + \)\(27\!\cdots\!47\)\( p^{44} T^{20} + \)\(29\!\cdots\!96\)\( p^{55} T^{21} + \)\(21\!\cdots\!46\)\( p^{66} T^{22} + \)\(14\!\cdots\!84\)\( p^{77} T^{23} + \)\(13\!\cdots\!78\)\( p^{88} T^{24} + \)\(52\!\cdots\!36\)\( p^{99} T^{25} + \)\(72\!\cdots\!58\)\( p^{110} T^{26} + \)\(98\!\cdots\!24\)\( p^{121} T^{27} + \)\(28\!\cdots\!13\)\( p^{132} T^{28} - \)\(43\!\cdots\!56\)\( p^{143} T^{29} + 75439096087978483550 p^{154} T^{30} - 511144020 p^{165} T^{31} + p^{176} T^{32} \)
59 \( 1 - 3728287296 T + \)\(25\!\cdots\!28\)\( T^{2} - \)\(64\!\cdots\!00\)\( T^{3} + \)\(30\!\cdots\!20\)\( T^{4} - \)\(47\!\cdots\!92\)\( T^{5} + \)\(24\!\cdots\!80\)\( T^{6} - \)\(14\!\cdots\!08\)\( T^{7} + \)\(15\!\cdots\!84\)\( T^{8} + \)\(42\!\cdots\!96\)\( T^{9} + \)\(74\!\cdots\!80\)\( T^{10} + \)\(75\!\cdots\!76\)\( T^{11} + \)\(31\!\cdots\!84\)\( T^{12} + \)\(47\!\cdots\!96\)\( T^{13} + \)\(11\!\cdots\!64\)\( T^{14} + \)\(19\!\cdots\!84\)\( T^{15} + \)\(35\!\cdots\!58\)\( T^{16} + \)\(19\!\cdots\!84\)\( p^{11} T^{17} + \)\(11\!\cdots\!64\)\( p^{22} T^{18} + \)\(47\!\cdots\!96\)\( p^{33} T^{19} + \)\(31\!\cdots\!84\)\( p^{44} T^{20} + \)\(75\!\cdots\!76\)\( p^{55} T^{21} + \)\(74\!\cdots\!80\)\( p^{66} T^{22} + \)\(42\!\cdots\!96\)\( p^{77} T^{23} + \)\(15\!\cdots\!84\)\( p^{88} T^{24} - \)\(14\!\cdots\!08\)\( p^{99} T^{25} + \)\(24\!\cdots\!80\)\( p^{110} T^{26} - \)\(47\!\cdots\!92\)\( p^{121} T^{27} + \)\(30\!\cdots\!20\)\( p^{132} T^{28} - \)\(64\!\cdots\!00\)\( p^{143} T^{29} + \)\(25\!\cdots\!28\)\( p^{154} T^{30} - 3728287296 p^{165} T^{31} + p^{176} T^{32} \)
61 \( 1 - 26227205052 T + \)\(72\!\cdots\!58\)\( T^{2} - \)\(13\!\cdots\!08\)\( T^{3} + \)\(22\!\cdots\!13\)\( T^{4} - \)\(30\!\cdots\!92\)\( T^{5} + \)\(40\!\cdots\!78\)\( T^{6} - \)\(46\!\cdots\!76\)\( T^{7} + \)\(82\!\cdots\!50\)\( p T^{8} - \)\(49\!\cdots\!88\)\( T^{9} + \)\(46\!\cdots\!30\)\( T^{10} - \)\(40\!\cdots\!40\)\( T^{11} + \)\(33\!\cdots\!71\)\( T^{12} - \)\(26\!\cdots\!28\)\( T^{13} + \)\(19\!\cdots\!50\)\( T^{14} - \)\(13\!\cdots\!84\)\( T^{15} + \)\(92\!\cdots\!22\)\( T^{16} - \)\(13\!\cdots\!84\)\( p^{11} T^{17} + \)\(19\!\cdots\!50\)\( p^{22} T^{18} - \)\(26\!\cdots\!28\)\( p^{33} T^{19} + \)\(33\!\cdots\!71\)\( p^{44} T^{20} - \)\(40\!\cdots\!40\)\( p^{55} T^{21} + \)\(46\!\cdots\!30\)\( p^{66} T^{22} - \)\(49\!\cdots\!88\)\( p^{77} T^{23} + \)\(82\!\cdots\!50\)\( p^{89} T^{24} - \)\(46\!\cdots\!76\)\( p^{99} T^{25} + \)\(40\!\cdots\!78\)\( p^{110} T^{26} - \)\(30\!\cdots\!92\)\( p^{121} T^{27} + \)\(22\!\cdots\!13\)\( p^{132} T^{28} - \)\(13\!\cdots\!08\)\( p^{143} T^{29} + \)\(72\!\cdots\!58\)\( p^{154} T^{30} - 26227205052 p^{165} T^{31} + p^{176} T^{32} \)
67 \( 1 + 5295680024 T + \)\(74\!\cdots\!28\)\( T^{2} + \)\(45\!\cdots\!48\)\( T^{3} + \)\(31\!\cdots\!36\)\( T^{4} + \)\(19\!\cdots\!56\)\( T^{5} + \)\(94\!\cdots\!04\)\( T^{6} + \)\(58\!\cdots\!40\)\( T^{7} + \)\(22\!\cdots\!28\)\( T^{8} + \)\(13\!\cdots\!44\)\( T^{9} + \)\(44\!\cdots\!08\)\( T^{10} + \)\(26\!\cdots\!04\)\( T^{11} + \)\(75\!\cdots\!56\)\( T^{12} + \)\(43\!\cdots\!96\)\( T^{13} + \)\(11\!\cdots\!96\)\( T^{14} + \)\(61\!\cdots\!52\)\( T^{15} + \)\(14\!\cdots\!82\)\( T^{16} + \)\(61\!\cdots\!52\)\( p^{11} T^{17} + \)\(11\!\cdots\!96\)\( p^{22} T^{18} + \)\(43\!\cdots\!96\)\( p^{33} T^{19} + \)\(75\!\cdots\!56\)\( p^{44} T^{20} + \)\(26\!\cdots\!04\)\( p^{55} T^{21} + \)\(44\!\cdots\!08\)\( p^{66} T^{22} + \)\(13\!\cdots\!44\)\( p^{77} T^{23} + \)\(22\!\cdots\!28\)\( p^{88} T^{24} + \)\(58\!\cdots\!40\)\( p^{99} T^{25} + \)\(94\!\cdots\!04\)\( p^{110} T^{26} + \)\(19\!\cdots\!56\)\( p^{121} T^{27} + \)\(31\!\cdots\!36\)\( p^{132} T^{28} + \)\(45\!\cdots\!48\)\( p^{143} T^{29} + \)\(74\!\cdots\!28\)\( p^{154} T^{30} + 5295680024 p^{165} T^{31} + p^{176} T^{32} \)
71 \( 1 - 17082800928 T + \)\(16\!\cdots\!84\)\( T^{2} - \)\(26\!\cdots\!36\)\( T^{3} + \)\(13\!\cdots\!72\)\( T^{4} - \)\(22\!\cdots\!12\)\( T^{5} + \)\(73\!\cdots\!72\)\( T^{6} - \)\(13\!\cdots\!96\)\( T^{7} + \)\(32\!\cdots\!56\)\( T^{8} - \)\(59\!\cdots\!56\)\( T^{9} + \)\(11\!\cdots\!56\)\( T^{10} - \)\(21\!\cdots\!52\)\( T^{11} + \)\(37\!\cdots\!00\)\( T^{12} - \)\(63\!\cdots\!60\)\( T^{13} + \)\(10\!\cdots\!76\)\( T^{14} - \)\(16\!\cdots\!52\)\( T^{15} + \)\(26\!\cdots\!78\)\( T^{16} - \)\(16\!\cdots\!52\)\( p^{11} T^{17} + \)\(10\!\cdots\!76\)\( p^{22} T^{18} - \)\(63\!\cdots\!60\)\( p^{33} T^{19} + \)\(37\!\cdots\!00\)\( p^{44} T^{20} - \)\(21\!\cdots\!52\)\( p^{55} T^{21} + \)\(11\!\cdots\!56\)\( p^{66} T^{22} - \)\(59\!\cdots\!56\)\( p^{77} T^{23} + \)\(32\!\cdots\!56\)\( p^{88} T^{24} - \)\(13\!\cdots\!96\)\( p^{99} T^{25} + \)\(73\!\cdots\!72\)\( p^{110} T^{26} - \)\(22\!\cdots\!12\)\( p^{121} T^{27} + \)\(13\!\cdots\!72\)\( p^{132} T^{28} - \)\(26\!\cdots\!36\)\( p^{143} T^{29} + \)\(16\!\cdots\!84\)\( p^{154} T^{30} - 17082800928 p^{165} T^{31} + p^{176} T^{32} \)
73 \( 1 - 21076301488 T + \)\(34\!\cdots\!80\)\( T^{2} - \)\(72\!\cdots\!48\)\( T^{3} + \)\(59\!\cdots\!50\)\( T^{4} - \)\(12\!\cdots\!92\)\( T^{5} + \)\(67\!\cdots\!20\)\( T^{6} - \)\(13\!\cdots\!44\)\( T^{7} + \)\(56\!\cdots\!29\)\( T^{8} - \)\(10\!\cdots\!16\)\( T^{9} + \)\(36\!\cdots\!52\)\( T^{10} - \)\(63\!\cdots\!12\)\( T^{11} + \)\(18\!\cdots\!18\)\( T^{12} - \)\(30\!\cdots\!76\)\( T^{13} + \)\(78\!\cdots\!36\)\( T^{14} - \)\(11\!\cdots\!44\)\( T^{15} + \)\(27\!\cdots\!52\)\( T^{16} - \)\(11\!\cdots\!44\)\( p^{11} T^{17} + \)\(78\!\cdots\!36\)\( p^{22} T^{18} - \)\(30\!\cdots\!76\)\( p^{33} T^{19} + \)\(18\!\cdots\!18\)\( p^{44} T^{20} - \)\(63\!\cdots\!12\)\( p^{55} T^{21} + \)\(36\!\cdots\!52\)\( p^{66} T^{22} - \)\(10\!\cdots\!16\)\( p^{77} T^{23} + \)\(56\!\cdots\!29\)\( p^{88} T^{24} - \)\(13\!\cdots\!44\)\( p^{99} T^{25} + \)\(67\!\cdots\!20\)\( p^{110} T^{26} - \)\(12\!\cdots\!92\)\( p^{121} T^{27} + \)\(59\!\cdots\!50\)\( p^{132} T^{28} - \)\(72\!\cdots\!48\)\( p^{143} T^{29} + \)\(34\!\cdots\!80\)\( p^{154} T^{30} - 21076301488 p^{165} T^{31} + p^{176} T^{32} \)
79 \( 1 + 83677977852 T + \)\(10\!\cdots\!06\)\( T^{2} + \)\(65\!\cdots\!68\)\( T^{3} + \)\(49\!\cdots\!65\)\( T^{4} + \)\(24\!\cdots\!24\)\( T^{5} + \)\(13\!\cdots\!38\)\( T^{6} + \)\(56\!\cdots\!84\)\( T^{7} + \)\(26\!\cdots\!42\)\( T^{8} + \)\(93\!\cdots\!92\)\( T^{9} + \)\(37\!\cdots\!54\)\( T^{10} + \)\(12\!\cdots\!84\)\( T^{11} + \)\(43\!\cdots\!47\)\( T^{12} + \)\(12\!\cdots\!96\)\( T^{13} + \)\(40\!\cdots\!38\)\( T^{14} + \)\(10\!\cdots\!12\)\( T^{15} + \)\(32\!\cdots\!38\)\( T^{16} + \)\(10\!\cdots\!12\)\( p^{11} T^{17} + \)\(40\!\cdots\!38\)\( p^{22} T^{18} + \)\(12\!\cdots\!96\)\( p^{33} T^{19} + \)\(43\!\cdots\!47\)\( p^{44} T^{20} + \)\(12\!\cdots\!84\)\( p^{55} T^{21} + \)\(37\!\cdots\!54\)\( p^{66} T^{22} + \)\(93\!\cdots\!92\)\( p^{77} T^{23} + \)\(26\!\cdots\!42\)\( p^{88} T^{24} + \)\(56\!\cdots\!84\)\( p^{99} T^{25} + \)\(13\!\cdots\!38\)\( p^{110} T^{26} + \)\(24\!\cdots\!24\)\( p^{121} T^{27} + \)\(49\!\cdots\!65\)\( p^{132} T^{28} + \)\(65\!\cdots\!68\)\( p^{143} T^{29} + \)\(10\!\cdots\!06\)\( p^{154} T^{30} + 83677977852 p^{165} T^{31} + p^{176} T^{32} \)
83 \( 1 + 72857072340 T + \)\(14\!\cdots\!18\)\( T^{2} + \)\(99\!\cdots\!44\)\( T^{3} + \)\(10\!\cdots\!97\)\( T^{4} + \)\(65\!\cdots\!04\)\( T^{5} + \)\(51\!\cdots\!66\)\( T^{6} + \)\(28\!\cdots\!96\)\( T^{7} + \)\(17\!\cdots\!70\)\( T^{8} + \)\(88\!\cdots\!52\)\( T^{9} + \)\(48\!\cdots\!78\)\( T^{10} + \)\(21\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!95\)\( T^{12} + \)\(41\!\cdots\!64\)\( T^{13} + \)\(17\!\cdots\!62\)\( T^{14} + \)\(64\!\cdots\!32\)\( T^{15} + \)\(25\!\cdots\!78\)\( T^{16} + \)\(64\!\cdots\!32\)\( p^{11} T^{17} + \)\(17\!\cdots\!62\)\( p^{22} T^{18} + \)\(41\!\cdots\!64\)\( p^{33} T^{19} + \)\(10\!\cdots\!95\)\( p^{44} T^{20} + \)\(21\!\cdots\!80\)\( p^{55} T^{21} + \)\(48\!\cdots\!78\)\( p^{66} T^{22} + \)\(88\!\cdots\!52\)\( p^{77} T^{23} + \)\(17\!\cdots\!70\)\( p^{88} T^{24} + \)\(28\!\cdots\!96\)\( p^{99} T^{25} + \)\(51\!\cdots\!66\)\( p^{110} T^{26} + \)\(65\!\cdots\!04\)\( p^{121} T^{27} + \)\(10\!\cdots\!97\)\( p^{132} T^{28} + \)\(99\!\cdots\!44\)\( p^{143} T^{29} + \)\(14\!\cdots\!18\)\( p^{154} T^{30} + 72857072340 p^{165} T^{31} + p^{176} T^{32} \)
89 \( 1 - 32238476676 T + \)\(21\!\cdots\!98\)\( T^{2} - \)\(83\!\cdots\!64\)\( T^{3} + \)\(24\!\cdots\!41\)\( T^{4} - \)\(10\!\cdots\!04\)\( T^{5} + \)\(18\!\cdots\!50\)\( T^{6} - \)\(88\!\cdots\!32\)\( T^{7} + \)\(11\!\cdots\!46\)\( T^{8} - \)\(55\!\cdots\!64\)\( T^{9} + \)\(51\!\cdots\!58\)\( T^{10} - \)\(27\!\cdots\!64\)\( T^{11} + \)\(20\!\cdots\!03\)\( T^{12} - \)\(10\!\cdots\!40\)\( T^{13} + \)\(67\!\cdots\!98\)\( T^{14} - \)\(36\!\cdots\!28\)\( T^{15} + \)\(20\!\cdots\!70\)\( T^{16} - \)\(36\!\cdots\!28\)\( p^{11} T^{17} + \)\(67\!\cdots\!98\)\( p^{22} T^{18} - \)\(10\!\cdots\!40\)\( p^{33} T^{19} + \)\(20\!\cdots\!03\)\( p^{44} T^{20} - \)\(27\!\cdots\!64\)\( p^{55} T^{21} + \)\(51\!\cdots\!58\)\( p^{66} T^{22} - \)\(55\!\cdots\!64\)\( p^{77} T^{23} + \)\(11\!\cdots\!46\)\( p^{88} T^{24} - \)\(88\!\cdots\!32\)\( p^{99} T^{25} + \)\(18\!\cdots\!50\)\( p^{110} T^{26} - \)\(10\!\cdots\!04\)\( p^{121} T^{27} + \)\(24\!\cdots\!41\)\( p^{132} T^{28} - \)\(83\!\cdots\!64\)\( p^{143} T^{29} + \)\(21\!\cdots\!98\)\( p^{154} T^{30} - 32238476676 p^{165} T^{31} + p^{176} T^{32} \)
97 \( 1 - 6973535140 T + \)\(60\!\cdots\!74\)\( T^{2} - \)\(15\!\cdots\!88\)\( T^{3} + \)\(18\!\cdots\!45\)\( T^{4} - \)\(69\!\cdots\!08\)\( T^{5} + \)\(37\!\cdots\!34\)\( T^{6} - \)\(16\!\cdots\!60\)\( T^{7} + \)\(59\!\cdots\!90\)\( T^{8} - \)\(26\!\cdots\!88\)\( T^{9} + \)\(74\!\cdots\!38\)\( T^{10} - \)\(33\!\cdots\!52\)\( T^{11} + \)\(78\!\cdots\!35\)\( T^{12} - \)\(33\!\cdots\!28\)\( T^{13} + \)\(70\!\cdots\!98\)\( T^{14} - \)\(28\!\cdots\!32\)\( T^{15} + \)\(54\!\cdots\!38\)\( T^{16} - \)\(28\!\cdots\!32\)\( p^{11} T^{17} + \)\(70\!\cdots\!98\)\( p^{22} T^{18} - \)\(33\!\cdots\!28\)\( p^{33} T^{19} + \)\(78\!\cdots\!35\)\( p^{44} T^{20} - \)\(33\!\cdots\!52\)\( p^{55} T^{21} + \)\(74\!\cdots\!38\)\( p^{66} T^{22} - \)\(26\!\cdots\!88\)\( p^{77} T^{23} + \)\(59\!\cdots\!90\)\( p^{88} T^{24} - \)\(16\!\cdots\!60\)\( p^{99} T^{25} + \)\(37\!\cdots\!34\)\( p^{110} T^{26} - \)\(69\!\cdots\!08\)\( p^{121} T^{27} + \)\(18\!\cdots\!45\)\( p^{132} T^{28} - \)\(15\!\cdots\!88\)\( p^{143} T^{29} + \)\(60\!\cdots\!74\)\( p^{154} T^{30} - 6973535140 p^{165} T^{31} + p^{176} T^{32} \)
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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

−2.24666920766488032945938277097, −2.06582430196036901970112772643, −2.01811109352503235110566679431, −1.94142832876732988022231595939, −1.89071680983690592381830824648, −1.88007101255774035696724298290, −1.78462523510510460654984329700, −1.73852548002558475071433947005, −1.69704781679607037733977066805, −1.67904639975996568713849948881, −1.67225266177460485655011513436, −1.53777056521559756902382925793, −1.32953941772206932151093335743, −1.24544456092908076262490881949, −1.17405744702200834885798427162, −1.16569841555585501120942483329, −1.07961153584622090385564780461, −0.949065967146797810703919260956, −0.945342638312064250656540050646, −0.933884221076162771319191758558, −0.872807889515845925953456750795, −0.795376565750484546912950739255, −0.75720425267183180441608037421, −0.74479967259017220814927949251, −0.68957815567530466681059461554, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.68957815567530466681059461554, 0.74479967259017220814927949251, 0.75720425267183180441608037421, 0.795376565750484546912950739255, 0.872807889515845925953456750795, 0.933884221076162771319191758558, 0.945342638312064250656540050646, 0.949065967146797810703919260956, 1.07961153584622090385564780461, 1.16569841555585501120942483329, 1.17405744702200834885798427162, 1.24544456092908076262490881949, 1.32953941772206932151093335743, 1.53777056521559756902382925793, 1.67225266177460485655011513436, 1.67904639975996568713849948881, 1.69704781679607037733977066805, 1.73852548002558475071433947005, 1.78462523510510460654984329700, 1.88007101255774035696724298290, 1.89071680983690592381830824648, 1.94142832876732988022231595939, 2.01811109352503235110566679431, 2.06582430196036901970112772643, 2.24666920766488032945938277097

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

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