Properties

Label 20-3e10-1.1-c36e10-0-0
Degree $20$
Conductor $59049$
Sign $1$
Analytic cond. $8.20694\times 10^{13}$
Root an. cond. $4.96259$
Motivic weight $36$
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  − 5.52e8·3-s + 1.25e11·4-s − 1.22e15·7-s + 1.87e17·9-s − 6.95e19·12-s + 1.46e20·13-s + 2.50e21·16-s + 2.51e23·19-s + 6.77e23·21-s + 6.91e25·25-s − 3.48e25·27-s − 1.54e26·28-s − 2.83e27·31-s + 2.35e28·36-s + 4.60e28·37-s − 8.08e28·39-s − 5.43e29·43-s − 1.38e30·48-s − 1.04e31·49-s + 1.84e31·52-s − 1.39e32·57-s + 2.50e32·61-s − 2.29e32·63-s − 3.61e32·64-s − 3.26e33·67-s − 4.35e33·73-s − 3.81e34·75-s + ⋯
L(s)  = 1  − 1.42·3-s + 1.83·4-s − 0.753·7-s + 1.24·9-s − 2.61·12-s + 1.30·13-s + 0.531·16-s + 2.41·19-s + 1.07·21-s + 4.75·25-s − 0.599·27-s − 1.38·28-s − 4.05·31-s + 2.28·36-s + 2.72·37-s − 1.85·39-s − 2.15·43-s − 0.757·48-s − 3.93·49-s + 2.38·52-s − 3.44·57-s + 1.83·61-s − 0.939·63-s − 1.11·64-s − 4.41·67-s − 1.25·73-s − 6.76·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+18)^{10} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(20\)
Conductor: \(59049\)    =    \(3^{10}\)
Sign: $1$
Analytic conductor: \(8.20694\times 10^{13}\)
Root analytic conductor: \(4.96259\)
Motivic weight: \(36\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((20,\ 59049,\ (\ :[18]^{10}),\ 1)\)

Particular Values

\(L(\frac{37}{2})\) \(\approx\) \(0.1159431498\)
\(L(\frac12)\) \(\approx\) \(0.1159431498\)
\(L(19)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 2272250 p^{5} T + 17950696265501 p^{8} T^{2} - 26667226960579400 p^{17} T^{3} - \)\(16\!\cdots\!48\)\( p^{30} T^{4} - \)\(61\!\cdots\!44\)\( p^{43} T^{5} - \)\(16\!\cdots\!48\)\( p^{66} T^{6} - 26667226960579400 p^{89} T^{7} + 17950696265501 p^{116} T^{8} + 2272250 p^{149} T^{9} + p^{180} T^{10} \)
good2 \( 1 - 15749654831 p^{3} T^{2} + 6526729578650332779 p^{11} T^{4} - \)\(23\!\cdots\!77\)\( p^{22} T^{6} + \)\(16\!\cdots\!29\)\( p^{39} T^{8} - \)\(20\!\cdots\!87\)\( p^{58} T^{10} + \)\(16\!\cdots\!29\)\( p^{111} T^{12} - \)\(23\!\cdots\!77\)\( p^{166} T^{14} + 6526729578650332779 p^{227} T^{16} - 15749654831 p^{291} T^{18} + p^{360} T^{20} \)
5 \( 1 - \)\(27\!\cdots\!46\)\( p^{2} T^{2} + \)\(77\!\cdots\!17\)\( p^{5} T^{4} - \)\(11\!\cdots\!08\)\( p^{11} T^{6} + \)\(11\!\cdots\!46\)\( p^{20} T^{8} - \)\(17\!\cdots\!44\)\( p^{30} T^{10} + \)\(11\!\cdots\!46\)\( p^{92} T^{12} - \)\(11\!\cdots\!08\)\( p^{155} T^{14} + \)\(77\!\cdots\!17\)\( p^{221} T^{16} - \)\(27\!\cdots\!46\)\( p^{290} T^{18} + p^{360} T^{20} \)
7 \( ( 1 + 87670922418050 p T + \)\(16\!\cdots\!47\)\( p^{3} T^{2} - \)\(16\!\cdots\!00\)\( p^{6} T^{3} + \)\(19\!\cdots\!58\)\( p^{8} T^{4} - \)\(86\!\cdots\!00\)\( p^{11} T^{5} + \)\(19\!\cdots\!58\)\( p^{44} T^{6} - \)\(16\!\cdots\!00\)\( p^{78} T^{7} + \)\(16\!\cdots\!47\)\( p^{111} T^{8} + 87670922418050 p^{145} T^{9} + p^{180} T^{10} )^{2} \)
11 \( 1 - \)\(17\!\cdots\!90\)\( p T^{2} + \)\(13\!\cdots\!35\)\( p^{3} T^{4} - \)\(58\!\cdots\!60\)\( p^{7} T^{6} + \)\(18\!\cdots\!90\)\( p^{11} T^{8} - \)\(36\!\cdots\!12\)\( p^{17} T^{10} + \)\(18\!\cdots\!90\)\( p^{83} T^{12} - \)\(58\!\cdots\!60\)\( p^{151} T^{14} + \)\(13\!\cdots\!35\)\( p^{219} T^{16} - \)\(17\!\cdots\!90\)\( p^{289} T^{18} + p^{360} T^{20} \)
13 \( ( 1 - 73221010691734315450 T + \)\(28\!\cdots\!37\)\( p T^{2} - \)\(21\!\cdots\!00\)\( p^{2} T^{3} + \)\(23\!\cdots\!78\)\( p^{4} T^{4} - \)\(14\!\cdots\!00\)\( p^{6} T^{5} + \)\(23\!\cdots\!78\)\( p^{40} T^{6} - \)\(21\!\cdots\!00\)\( p^{74} T^{7} + \)\(28\!\cdots\!37\)\( p^{109} T^{8} - 73221010691734315450 p^{144} T^{9} + p^{180} T^{10} )^{2} \)
17 \( 1 - \)\(43\!\cdots\!62\)\( p^{2} T^{2} + \)\(96\!\cdots\!97\)\( p^{4} T^{4} - \)\(14\!\cdots\!72\)\( p^{6} T^{6} + \)\(51\!\cdots\!78\)\( p^{10} T^{8} - \)\(13\!\cdots\!12\)\( p^{14} T^{10} + \)\(51\!\cdots\!78\)\( p^{82} T^{12} - \)\(14\!\cdots\!72\)\( p^{150} T^{14} + \)\(96\!\cdots\!97\)\( p^{220} T^{16} - \)\(43\!\cdots\!62\)\( p^{290} T^{18} + p^{360} T^{20} \)
19 \( ( 1 - \)\(12\!\cdots\!18\)\( T + \)\(33\!\cdots\!97\)\( T^{2} - \)\(23\!\cdots\!52\)\( p T^{3} + \)\(17\!\cdots\!62\)\( p^{2} T^{4} - \)\(95\!\cdots\!72\)\( p^{3} T^{5} + \)\(17\!\cdots\!62\)\( p^{38} T^{6} - \)\(23\!\cdots\!52\)\( p^{73} T^{7} + \)\(33\!\cdots\!97\)\( p^{108} T^{8} - \)\(12\!\cdots\!18\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
23 \( 1 - \)\(44\!\cdots\!18\)\( T^{2} + \)\(12\!\cdots\!17\)\( T^{4} - \)\(44\!\cdots\!32\)\( p^{2} T^{6} + \)\(12\!\cdots\!22\)\( p^{4} T^{8} - \)\(52\!\cdots\!08\)\( p^{8} T^{10} + \)\(12\!\cdots\!22\)\( p^{76} T^{12} - \)\(44\!\cdots\!32\)\( p^{146} T^{14} + \)\(12\!\cdots\!17\)\( p^{216} T^{16} - \)\(44\!\cdots\!18\)\( p^{288} T^{18} + p^{360} T^{20} \)
29 \( 1 - \)\(15\!\cdots\!30\)\( T^{2} + \)\(16\!\cdots\!05\)\( p^{2} T^{4} - \)\(44\!\cdots\!20\)\( p^{5} T^{6} + \)\(86\!\cdots\!30\)\( p^{6} T^{8} - \)\(49\!\cdots\!32\)\( p^{8} T^{10} + \)\(86\!\cdots\!30\)\( p^{78} T^{12} - \)\(44\!\cdots\!20\)\( p^{149} T^{14} + \)\(16\!\cdots\!05\)\( p^{218} T^{16} - \)\(15\!\cdots\!30\)\( p^{288} T^{18} + p^{360} T^{20} \)
31 \( ( 1 + \)\(45\!\cdots\!10\)\( p T + \)\(21\!\cdots\!85\)\( p^{2} T^{2} + \)\(65\!\cdots\!40\)\( p^{3} T^{3} + \)\(18\!\cdots\!90\)\( p^{4} T^{4} + \)\(43\!\cdots\!48\)\( p^{5} T^{5} + \)\(18\!\cdots\!90\)\( p^{40} T^{6} + \)\(65\!\cdots\!40\)\( p^{75} T^{7} + \)\(21\!\cdots\!85\)\( p^{110} T^{8} + \)\(45\!\cdots\!10\)\( p^{145} T^{9} + p^{180} T^{10} )^{2} \)
37 \( ( 1 - \)\(23\!\cdots\!50\)\( T + \)\(98\!\cdots\!81\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(45\!\cdots\!58\)\( T^{4} - \)\(82\!\cdots\!00\)\( T^{5} + \)\(45\!\cdots\!58\)\( p^{36} T^{6} - \)\(21\!\cdots\!00\)\( p^{72} T^{7} + \)\(98\!\cdots\!81\)\( p^{108} T^{8} - \)\(23\!\cdots\!50\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
41 \( 1 - \)\(23\!\cdots\!90\)\( T^{2} + \)\(10\!\cdots\!85\)\( T^{4} + \)\(58\!\cdots\!40\)\( T^{6} + \)\(19\!\cdots\!90\)\( T^{8} - \)\(48\!\cdots\!52\)\( T^{10} + \)\(19\!\cdots\!90\)\( p^{72} T^{12} + \)\(58\!\cdots\!40\)\( p^{144} T^{14} + \)\(10\!\cdots\!85\)\( p^{216} T^{16} - \)\(23\!\cdots\!90\)\( p^{288} T^{18} + p^{360} T^{20} \)
43 \( ( 1 + \)\(27\!\cdots\!50\)\( T + \)\(27\!\cdots\!81\)\( T^{2} + \)\(57\!\cdots\!00\)\( T^{3} + \)\(31\!\cdots\!38\)\( T^{4} + \)\(50\!\cdots\!00\)\( T^{5} + \)\(31\!\cdots\!38\)\( p^{36} T^{6} + \)\(57\!\cdots\!00\)\( p^{72} T^{7} + \)\(27\!\cdots\!81\)\( p^{108} T^{8} + \)\(27\!\cdots\!50\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
47 \( 1 - \)\(90\!\cdots\!18\)\( T^{2} + \)\(34\!\cdots\!97\)\( T^{4} - \)\(16\!\cdots\!64\)\( p T^{6} + \)\(10\!\cdots\!42\)\( T^{8} - \)\(14\!\cdots\!28\)\( T^{10} + \)\(10\!\cdots\!42\)\( p^{72} T^{12} - \)\(16\!\cdots\!64\)\( p^{145} T^{14} + \)\(34\!\cdots\!97\)\( p^{216} T^{16} - \)\(90\!\cdots\!18\)\( p^{288} T^{18} + p^{360} T^{20} \)
53 \( 1 - \)\(62\!\cdots\!18\)\( T^{2} + \)\(19\!\cdots\!97\)\( T^{4} - \)\(38\!\cdots\!08\)\( T^{6} + \)\(57\!\cdots\!42\)\( T^{8} - \)\(72\!\cdots\!28\)\( T^{10} + \)\(57\!\cdots\!42\)\( p^{72} T^{12} - \)\(38\!\cdots\!08\)\( p^{144} T^{14} + \)\(19\!\cdots\!97\)\( p^{216} T^{16} - \)\(62\!\cdots\!18\)\( p^{288} T^{18} + p^{360} T^{20} \)
59 \( 1 - \)\(26\!\cdots\!90\)\( T^{2} + \)\(39\!\cdots\!85\)\( T^{4} - \)\(40\!\cdots\!60\)\( T^{6} + \)\(31\!\cdots\!90\)\( T^{8} - \)\(19\!\cdots\!52\)\( T^{10} + \)\(31\!\cdots\!90\)\( p^{72} T^{12} - \)\(40\!\cdots\!60\)\( p^{144} T^{14} + \)\(39\!\cdots\!85\)\( p^{216} T^{16} - \)\(26\!\cdots\!90\)\( p^{288} T^{18} + p^{360} T^{20} \)
61 \( ( 1 - \)\(12\!\cdots\!10\)\( T + \)\(40\!\cdots\!45\)\( T^{2} - \)\(51\!\cdots\!20\)\( T^{3} + \)\(12\!\cdots\!10\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(12\!\cdots\!10\)\( p^{36} T^{6} - \)\(51\!\cdots\!20\)\( p^{72} T^{7} + \)\(40\!\cdots\!45\)\( p^{108} T^{8} - \)\(12\!\cdots\!10\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
67 \( ( 1 + \)\(16\!\cdots\!50\)\( T + \)\(32\!\cdots\!41\)\( T^{2} + \)\(34\!\cdots\!00\)\( T^{3} + \)\(38\!\cdots\!58\)\( T^{4} + \)\(27\!\cdots\!00\)\( T^{5} + \)\(38\!\cdots\!58\)\( p^{36} T^{6} + \)\(34\!\cdots\!00\)\( p^{72} T^{7} + \)\(32\!\cdots\!41\)\( p^{108} T^{8} + \)\(16\!\cdots\!50\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
71 \( 1 - \)\(31\!\cdots\!30\)\( T^{2} + \)\(49\!\cdots\!05\)\( T^{4} - \)\(96\!\cdots\!80\)\( p^{2} T^{6} + \)\(13\!\cdots\!30\)\( p^{4} T^{8} - \)\(13\!\cdots\!12\)\( p^{6} T^{10} + \)\(13\!\cdots\!30\)\( p^{76} T^{12} - \)\(96\!\cdots\!80\)\( p^{146} T^{14} + \)\(49\!\cdots\!05\)\( p^{216} T^{16} - \)\(31\!\cdots\!30\)\( p^{288} T^{18} + p^{360} T^{20} \)
73 \( ( 1 + \)\(21\!\cdots\!50\)\( T + \)\(41\!\cdots\!41\)\( T^{2} + \)\(83\!\cdots\!00\)\( T^{3} + \)\(84\!\cdots\!98\)\( T^{4} + \)\(13\!\cdots\!00\)\( T^{5} + \)\(84\!\cdots\!98\)\( p^{36} T^{6} + \)\(83\!\cdots\!00\)\( p^{72} T^{7} + \)\(41\!\cdots\!41\)\( p^{108} T^{8} + \)\(21\!\cdots\!50\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
79 \( ( 1 + \)\(31\!\cdots\!42\)\( T + \)\(86\!\cdots\!57\)\( T^{2} + \)\(21\!\cdots\!28\)\( p T^{3} + \)\(29\!\cdots\!62\)\( T^{4} + \)\(42\!\cdots\!52\)\( T^{5} + \)\(29\!\cdots\!62\)\( p^{36} T^{6} + \)\(21\!\cdots\!28\)\( p^{73} T^{7} + \)\(86\!\cdots\!57\)\( p^{108} T^{8} + \)\(31\!\cdots\!42\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
83 \( 1 - \)\(70\!\cdots\!98\)\( T^{2} + \)\(26\!\cdots\!37\)\( T^{4} - \)\(64\!\cdots\!48\)\( T^{6} + \)\(11\!\cdots\!22\)\( T^{8} - \)\(16\!\cdots\!28\)\( T^{10} + \)\(11\!\cdots\!22\)\( p^{72} T^{12} - \)\(64\!\cdots\!48\)\( p^{144} T^{14} + \)\(26\!\cdots\!37\)\( p^{216} T^{16} - \)\(70\!\cdots\!98\)\( p^{288} T^{18} + p^{360} T^{20} \)
89 \( 1 - \)\(39\!\cdots\!90\)\( T^{2} + \)\(94\!\cdots\!85\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(40\!\cdots\!90\)\( T^{8} - \)\(61\!\cdots\!52\)\( T^{10} + \)\(40\!\cdots\!90\)\( p^{72} T^{12} - \)\(22\!\cdots\!60\)\( p^{144} T^{14} + \)\(94\!\cdots\!85\)\( p^{216} T^{16} - \)\(39\!\cdots\!90\)\( p^{288} T^{18} + p^{360} T^{20} \)
97 \( ( 1 + \)\(25\!\cdots\!50\)\( T + \)\(11\!\cdots\!81\)\( T^{2} + \)\(43\!\cdots\!00\)\( T^{3} + \)\(60\!\cdots\!98\)\( T^{4} + \)\(23\!\cdots\!00\)\( T^{5} + \)\(60\!\cdots\!98\)\( p^{36} T^{6} + \)\(43\!\cdots\!00\)\( p^{72} T^{7} + \)\(11\!\cdots\!81\)\( p^{108} T^{8} + \)\(25\!\cdots\!50\)\( p^{144} T^{9} + p^{180} T^{10} )^{2} \)
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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

−4.75557280413119842863514083162, −4.75242817224134003750730303875, −4.63211561198509072678070456405, −4.14471201839254364311439302915, −3.99235354564212117553334177222, −3.78251511989847344502192631033, −3.47383673786197691960977962242, −3.30933037888955014482871279920, −3.09616954905750847359994087091, −2.94698797465139545560106051261, −2.91412371371228484820966172743, −2.76870862969677278276340936532, −2.73301557727581624006663229202, −2.02687691957045215412025742080, −1.96188350160833035899264126579, −1.68259050087227049599615018847, −1.61851187774991455676441379273, −1.50188740415607498250490674151, −1.43750552745232256077835493214, −1.08991605284381146916771936616, −0.74560183464791159840684000864, −0.72638315059615559206983364755, −0.52958644252986728793728414214, −0.44551898742385936365824221444, −0.01643414604380455014021311077, 0.01643414604380455014021311077, 0.44551898742385936365824221444, 0.52958644252986728793728414214, 0.72638315059615559206983364755, 0.74560183464791159840684000864, 1.08991605284381146916771936616, 1.43750552745232256077835493214, 1.50188740415607498250490674151, 1.61851187774991455676441379273, 1.68259050087227049599615018847, 1.96188350160833035899264126579, 2.02687691957045215412025742080, 2.73301557727581624006663229202, 2.76870862969677278276340936532, 2.91412371371228484820966172743, 2.94698797465139545560106051261, 3.09616954905750847359994087091, 3.30933037888955014482871279920, 3.47383673786197691960977962242, 3.78251511989847344502192631033, 3.99235354564212117553334177222, 4.14471201839254364311439302915, 4.63211561198509072678070456405, 4.75242817224134003750730303875, 4.75557280413119842863514083162

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

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