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Properties

Label 6-24e3-1.1-c21e3-0-0
Degree $6$
Conductor $13824$
Sign $1$
Analytic cond. $301768.$
Root an. cond. $8.18990$
Motivic weight $21$
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  − 1.77e5·3-s − 4.83e6·5-s + 2.71e8·7-s + 2.09e10·9-s + 6.11e10·11-s − 5.94e11·13-s + 8.56e11·15-s + 1.42e12·17-s − 3.50e13·19-s − 4.80e13·21-s − 9.49e13·23-s − 1.01e15·25-s − 2.05e15·27-s − 3.29e15·29-s − 1.92e15·31-s − 1.08e16·33-s − 1.31e15·35-s + 2.21e16·37-s + 1.05e17·39-s + 1.67e17·41-s + 6.95e16·43-s − 1.01e17·45-s − 4.91e16·47-s − 3.14e17·49-s − 2.52e17·51-s + 4.88e17·53-s − 2.95e17·55-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.221·5-s + 0.363·7-s + 2·9-s + 0.711·11-s − 1.19·13-s + 0.383·15-s + 0.171·17-s − 1.31·19-s − 0.629·21-s − 0.477·23-s − 2.11·25-s − 1.92·27-s − 1.45·29-s − 0.422·31-s − 1.23·33-s − 0.0803·35-s + 0.758·37-s + 2.07·39-s + 1.94·41-s + 0.490·43-s − 0.442·45-s − 0.136·47-s − 0.562·49-s − 0.296·51-s + 0.383·53-s − 0.157·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13824 ^{s/2} \, \Gamma_{\C}(s+21/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(13824\)    =    \(2^{9} \cdot 3^{3}\)
Sign: $1$
Analytic conductor: \(301768.\)
Root analytic conductor: \(8.18990\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 13824,\ (\ :21/2, 21/2, 21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(1.743921966\)
\(L(\frac12)\) \(\approx\) \(1.743921966\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2 \( 1 \)
3$C_1$ \( ( 1 + p^{10} T )^{3} \)
good5$S_4\times C_2$ \( 1 + 4833126 T + 41353012136739 p^{2} T^{2} + 7997585659247680772 p^{4} T^{3} + 41353012136739 p^{23} T^{4} + 4833126 p^{42} T^{5} + p^{63} T^{6} \)
7$S_4\times C_2$ \( 1 - 271431024 T + 55427973478273443 p T^{2} + \)\(40\!\cdots\!36\)\( p^{2} T^{3} + 55427973478273443 p^{22} T^{4} - 271431024 p^{42} T^{5} + p^{63} T^{6} \)
11$S_4\times C_2$ \( 1 - 61194658188 T + \)\(14\!\cdots\!79\)\( p T^{2} - \)\(82\!\cdots\!16\)\( p^{2} T^{3} + \)\(14\!\cdots\!79\)\( p^{22} T^{4} - 61194658188 p^{42} T^{5} + p^{63} T^{6} \)
13$S_4\times C_2$ \( 1 + 594486202422 T - \)\(28\!\cdots\!77\)\( p T^{2} - \)\(10\!\cdots\!76\)\( p^{2} T^{3} - \)\(28\!\cdots\!77\)\( p^{22} T^{4} + 594486202422 p^{42} T^{5} + p^{63} T^{6} \)
17$S_4\times C_2$ \( 1 - 83812912902 p T + \)\(34\!\cdots\!27\)\( p^{2} T^{2} - \)\(73\!\cdots\!16\)\( p^{3} T^{3} + \)\(34\!\cdots\!27\)\( p^{23} T^{4} - 83812912902 p^{43} T^{5} + p^{63} T^{6} \)
19$S_4\times C_2$ \( 1 + 1847096088516 p T + \)\(32\!\cdots\!81\)\( p^{2} T^{2} + \)\(38\!\cdots\!32\)\( p^{3} T^{3} + \)\(32\!\cdots\!81\)\( p^{23} T^{4} + 1847096088516 p^{43} T^{5} + p^{63} T^{6} \)
23$S_4\times C_2$ \( 1 + 94911043073592 T - \)\(36\!\cdots\!95\)\( T^{2} - \)\(12\!\cdots\!56\)\( T^{3} - \)\(36\!\cdots\!95\)\( p^{21} T^{4} + 94911043073592 p^{42} T^{5} + p^{63} T^{6} \)
29$S_4\times C_2$ \( 1 + 3292734981070734 T + \)\(11\!\cdots\!51\)\( T^{2} + \)\(30\!\cdots\!48\)\( T^{3} + \)\(11\!\cdots\!51\)\( p^{21} T^{4} + 3292734981070734 p^{42} T^{5} + p^{63} T^{6} \)
31$S_4\times C_2$ \( 1 + 1928658139449144 T + \)\(41\!\cdots\!77\)\( T^{2} + \)\(40\!\cdots\!28\)\( T^{3} + \)\(41\!\cdots\!77\)\( p^{21} T^{4} + 1928658139449144 p^{42} T^{5} + p^{63} T^{6} \)
37$S_4\times C_2$ \( 1 - 22185275061682722 T - \)\(46\!\cdots\!53\)\( p T^{2} + \)\(36\!\cdots\!48\)\( T^{3} - \)\(46\!\cdots\!53\)\( p^{22} T^{4} - 22185275061682722 p^{42} T^{5} + p^{63} T^{6} \)
41$S_4\times C_2$ \( 1 - 167394191883971118 T + \)\(26\!\cdots\!19\)\( T^{2} - \)\(24\!\cdots\!36\)\( T^{3} + \)\(26\!\cdots\!19\)\( p^{21} T^{4} - 167394191883971118 p^{42} T^{5} + p^{63} T^{6} \)
43$S_4\times C_2$ \( 1 - 69547615491362508 T + \)\(58\!\cdots\!17\)\( T^{2} - \)\(27\!\cdots\!44\)\( T^{3} + \)\(58\!\cdots\!17\)\( p^{21} T^{4} - 69547615491362508 p^{42} T^{5} + p^{63} T^{6} \)
47$S_4\times C_2$ \( 1 + 49110618973894752 T + \)\(29\!\cdots\!61\)\( T^{2} + \)\(26\!\cdots\!88\)\( T^{3} + \)\(29\!\cdots\!61\)\( p^{21} T^{4} + 49110618973894752 p^{42} T^{5} + p^{63} T^{6} \)
53$S_4\times C_2$ \( 1 - 488391341947078074 T + \)\(26\!\cdots\!23\)\( T^{2} - \)\(21\!\cdots\!84\)\( T^{3} + \)\(26\!\cdots\!23\)\( p^{21} T^{4} - 488391341947078074 p^{42} T^{5} + p^{63} T^{6} \)
59$S_4\times C_2$ \( 1 + 1846144806145488852 T + \)\(30\!\cdots\!53\)\( T^{2} + \)\(22\!\cdots\!36\)\( T^{3} + \)\(30\!\cdots\!53\)\( p^{21} T^{4} + 1846144806145488852 p^{42} T^{5} + p^{63} T^{6} \)
61$S_4\times C_2$ \( 1 - 8183707455377473002 T + \)\(11\!\cdots\!63\)\( T^{2} - \)\(52\!\cdots\!68\)\( T^{3} + \)\(11\!\cdots\!63\)\( p^{21} T^{4} - 8183707455377473002 p^{42} T^{5} + p^{63} T^{6} \)
67$S_4\times C_2$ \( 1 - 37438935583939030164 T + \)\(94\!\cdots\!85\)\( T^{2} - \)\(16\!\cdots\!52\)\( T^{3} + \)\(94\!\cdots\!85\)\( p^{21} T^{4} - 37438935583939030164 p^{42} T^{5} + p^{63} T^{6} \)
71$S_4\times C_2$ \( 1 - 65865243803437791096 T + \)\(35\!\cdots\!17\)\( T^{2} - \)\(10\!\cdots\!32\)\( T^{3} + \)\(35\!\cdots\!17\)\( p^{21} T^{4} - 65865243803437791096 p^{42} T^{5} + p^{63} T^{6} \)
73$S_4\times C_2$ \( 1 - 88415465494171854846 T + \)\(56\!\cdots\!79\)\( T^{2} - \)\(22\!\cdots\!16\)\( T^{3} + \)\(56\!\cdots\!79\)\( p^{21} T^{4} - 88415465494171854846 p^{42} T^{5} + p^{63} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(22\!\cdots\!20\)\( T + \)\(30\!\cdots\!05\)\( T^{2} - \)\(28\!\cdots\!24\)\( T^{3} + \)\(30\!\cdots\!05\)\( p^{21} T^{4} - \)\(22\!\cdots\!20\)\( p^{42} T^{5} + p^{63} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(84\!\cdots\!41\)\( T^{2} - \)\(12\!\cdots\!32\)\( T^{3} + \)\(84\!\cdots\!41\)\( p^{21} T^{4} - \)\(30\!\cdots\!00\)\( p^{42} T^{5} + p^{63} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(32\!\cdots\!70\)\( T + \)\(25\!\cdots\!59\)\( T^{2} - \)\(53\!\cdots\!72\)\( T^{3} + \)\(25\!\cdots\!59\)\( p^{21} T^{4} - \)\(32\!\cdots\!70\)\( p^{42} T^{5} + p^{63} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(19\!\cdots\!58\)\( T + \)\(23\!\cdots\!87\)\( T^{2} - \)\(18\!\cdots\!20\)\( T^{3} + \)\(23\!\cdots\!87\)\( p^{21} T^{4} - \)\(19\!\cdots\!58\)\( p^{42} T^{5} + p^{63} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.52117424448509037553785709732, −11.06642001710190542814461418779, −10.65992986271534509123429846952, −10.24338680492620443619783518841, −9.658771687815691641308325116382, −9.345953172582984244187821430346, −9.111725344407313911060322781837, −7.88336499545241624567003532619, −7.80611933764111764755755168186, −7.68567444336313273617687283054, −6.66873099847123287985689117626, −6.44610947385001636832652884213, −6.24608792439763858787688831017, −5.38443255105812889436696642285, −5.31164272529152100670736742182, −4.83515674906548987224298604168, −4.11904464970688657722884018918, −3.81489739645863349553177063322, −3.68445939838729216911761680481, −2.30070316052647863795086609749, −2.04432375672692400343768485538, −1.98366446922254541381736427766, −0.885308499392363757435002957724, −0.64387260835173159331991917253, −0.36760358792230465215597830610, 0.36760358792230465215597830610, 0.64387260835173159331991917253, 0.885308499392363757435002957724, 1.98366446922254541381736427766, 2.04432375672692400343768485538, 2.30070316052647863795086609749, 3.68445939838729216911761680481, 3.81489739645863349553177063322, 4.11904464970688657722884018918, 4.83515674906548987224298604168, 5.31164272529152100670736742182, 5.38443255105812889436696642285, 6.24608792439763858787688831017, 6.44610947385001636832652884213, 6.66873099847123287985689117626, 7.68567444336313273617687283054, 7.80611933764111764755755168186, 7.88336499545241624567003532619, 9.111725344407313911060322781837, 9.345953172582984244187821430346, 9.658771687815691641308325116382, 10.24338680492620443619783518841, 10.65992986271534509123429846952, 11.06642001710190542814461418779, 11.52117424448509037553785709732

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