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Properties

Label 6-11e3-1.1-c11e3-0-0
Degree $6$
Conductor $1331$
Sign $-1$
Analytic cond. $603.731$
Root an. cond. $2.90719$
Motivic weight $11$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 393·3-s − 1.64e3·4-s − 7.30e3·5-s − 5.08e3·7-s − 9.22e4·8-s − 2.87e5·9-s + 4.83e5·11-s + 6.46e5·12-s − 2.43e6·13-s + 2.87e6·15-s − 6.64e5·16-s + 1.21e7·17-s − 8.59e6·19-s + 1.20e7·20-s + 1.99e6·21-s − 3.13e6·23-s + 3.62e7·24-s − 1.74e7·25-s + 1.81e8·27-s + 8.35e6·28-s − 3.76e8·29-s − 3.13e8·31-s + 3.03e8·32-s − 1.89e8·33-s + 3.71e7·35-s + 4.72e8·36-s − 4.54e8·37-s + ⋯
L(s)  = 1  − 0.933·3-s − 0.802·4-s − 1.04·5-s − 0.114·7-s − 0.995·8-s − 1.62·9-s + 0.904·11-s + 0.749·12-s − 1.81·13-s + 0.976·15-s − 0.158·16-s + 2.06·17-s − 0.795·19-s + 0.839·20-s + 0.106·21-s − 0.101·23-s + 0.929·24-s − 0.358·25-s + 2.43·27-s + 0.0917·28-s − 3.40·29-s − 1.96·31-s + 1.59·32-s − 0.844·33-s + 0.119·35-s + 1.30·36-s − 1.07·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+11/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(1331\)    =    \(11^{3}\)
Sign: $-1$
Analytic conductor: \(603.731\)
Root analytic conductor: \(2.90719\)
Motivic weight: \(11\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 1331,\ (\ :11/2, 11/2, 11/2),\ -1)\)

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$$\Gal(F_p)$$F_p(T)$
bad11$C_1$ \( ( 1 - p^{5} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 411 p^{2} T^{2} + 1441 p^{6} T^{3} + 411 p^{13} T^{4} + p^{33} T^{6} \)
3$S_4\times C_2$ \( 1 + 131 p T + 16372 p^{3} T^{2} + 433067 p^{5} T^{3} + 16372 p^{14} T^{4} + 131 p^{23} T^{5} + p^{33} T^{6} \)
5$S_4\times C_2$ \( 1 + 1461 p T + 2833794 p^{2} T^{2} + 6204977609 p^{3} T^{3} + 2833794 p^{13} T^{4} + 1461 p^{23} T^{5} + p^{33} T^{6} \)
7$S_4\times C_2$ \( 1 + 726 p T + 481723191 p T^{2} + 63156675860036 T^{3} + 481723191 p^{12} T^{4} + 726 p^{23} T^{5} + p^{33} T^{6} \)
13$S_4\times C_2$ \( 1 + 2434212 T + 6882942919935 T^{2} + 8758917974666663144 T^{3} + 6882942919935 p^{11} T^{4} + 2434212 p^{22} T^{5} + p^{33} T^{6} \)
17$S_4\times C_2$ \( 1 - 12112122 T + 131101561369167 T^{2} - \)\(80\!\cdots\!04\)\( T^{3} + 131101561369167 p^{11} T^{4} - 12112122 p^{22} T^{5} + p^{33} T^{6} \)
19$S_4\times C_2$ \( 1 + 8590560 T + 144459522084657 T^{2} + \)\(34\!\cdots\!80\)\( T^{3} + 144459522084657 p^{11} T^{4} + 8590560 p^{22} T^{5} + p^{33} T^{6} \)
23$S_4\times C_2$ \( 1 + 3136413 T + 1080204993811464 T^{2} + \)\(20\!\cdots\!61\)\( T^{3} + 1080204993811464 p^{11} T^{4} + 3136413 p^{22} T^{5} + p^{33} T^{6} \)
29$S_4\times C_2$ \( 1 + 376441824 T + 82530722143886319 T^{2} + \)\(11\!\cdots\!32\)\( T^{3} + 82530722143886319 p^{11} T^{4} + 376441824 p^{22} T^{5} + p^{33} T^{6} \)
31$S_4\times C_2$ \( 1 + 313174893 T + 103090007115016368 T^{2} + \)\(16\!\cdots\!41\)\( T^{3} + 103090007115016368 p^{11} T^{4} + 313174893 p^{22} T^{5} + p^{33} T^{6} \)
37$S_4\times C_2$ \( 1 + 454281387 T + 581699696111812554 T^{2} + \)\(16\!\cdots\!39\)\( T^{3} + 581699696111812554 p^{11} T^{4} + 454281387 p^{22} T^{5} + p^{33} T^{6} \)
41$S_4\times C_2$ \( 1 + 37614456 T + 4963957689080163 p T^{2} + \)\(64\!\cdots\!00\)\( T^{3} + 4963957689080163 p^{12} T^{4} + 37614456 p^{22} T^{5} + p^{33} T^{6} \)
43$S_4\times C_2$ \( 1 - 162163386 T + 2181606843430351653 T^{2} - \)\(23\!\cdots\!24\)\( p T^{3} + 2181606843430351653 p^{11} T^{4} - 162163386 p^{22} T^{5} + p^{33} T^{6} \)
47$S_4\times C_2$ \( 1 + 3182498184 T + 7470690863276257197 T^{2} + \)\(11\!\cdots\!76\)\( T^{3} + 7470690863276257197 p^{11} T^{4} + 3182498184 p^{22} T^{5} + p^{33} T^{6} \)
53$S_4\times C_2$ \( 1 + 3000753402 T + 29214069482162931219 T^{2} + \)\(55\!\cdots\!16\)\( T^{3} + 29214069482162931219 p^{11} T^{4} + 3000753402 p^{22} T^{5} + p^{33} T^{6} \)
59$S_4\times C_2$ \( 1 + 1843219707 T + 61192633279808684556 T^{2} + \)\(84\!\cdots\!11\)\( T^{3} + 61192633279808684556 p^{11} T^{4} + 1843219707 p^{22} T^{5} + p^{33} T^{6} \)
61$S_4\times C_2$ \( 1 + 28094112684 T + \)\(39\!\cdots\!67\)\( T^{2} + \)\(32\!\cdots\!16\)\( T^{3} + \)\(39\!\cdots\!67\)\( p^{11} T^{4} + 28094112684 p^{22} T^{5} + p^{33} T^{6} \)
67$S_4\times C_2$ \( 1 - 10315312497 T + \)\(16\!\cdots\!96\)\( T^{2} - \)\(16\!\cdots\!93\)\( T^{3} + \)\(16\!\cdots\!96\)\( p^{11} T^{4} - 10315312497 p^{22} T^{5} + p^{33} T^{6} \)
71$S_4\times C_2$ \( 1 - 3703071657 T + \)\(20\!\cdots\!20\)\( T^{2} - \)\(28\!\cdots\!33\)\( T^{3} + \)\(20\!\cdots\!20\)\( p^{11} T^{4} - 3703071657 p^{22} T^{5} + p^{33} T^{6} \)
73$S_4\times C_2$ \( 1 - 14017034988 T + \)\(62\!\cdots\!75\)\( T^{2} - \)\(82\!\cdots\!76\)\( T^{3} + \)\(62\!\cdots\!75\)\( p^{11} T^{4} - 14017034988 p^{22} T^{5} + p^{33} T^{6} \)
79$S_4\times C_2$ \( 1 + 8104583058 T + \)\(61\!\cdots\!33\)\( T^{2} + \)\(32\!\cdots\!84\)\( T^{3} + \)\(61\!\cdots\!33\)\( p^{11} T^{4} + 8104583058 p^{22} T^{5} + p^{33} T^{6} \)
83$S_4\times C_2$ \( 1 - 26009027946 T + \)\(62\!\cdots\!73\)\( T^{2} - \)\(12\!\cdots\!32\)\( T^{3} + \)\(62\!\cdots\!73\)\( p^{11} T^{4} - 26009027946 p^{22} T^{5} + p^{33} T^{6} \)
89$S_4\times C_2$ \( 1 + 17344395051 T + \)\(18\!\cdots\!38\)\( T^{2} - \)\(12\!\cdots\!37\)\( T^{3} + \)\(18\!\cdots\!38\)\( p^{11} T^{4} + 17344395051 p^{22} T^{5} + p^{33} T^{6} \)
97$S_4\times C_2$ \( 1 + 7984545237 T + \)\(19\!\cdots\!74\)\( T^{2} + \)\(85\!\cdots\!29\)\( T^{3} + \)\(19\!\cdots\!74\)\( p^{11} T^{4} + 7984545237 p^{22} T^{5} + p^{33} T^{6} \)
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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

−16.86808027554863387250684260478, −16.51957543336793069605193661058, −15.59945551870535723137625894146, −14.93257418061147758179368688434, −14.63051300288929024087883648916, −14.46065095183698922040384545625, −13.89908393504326071701518772159, −12.69135701513639340876031820186, −12.65337726312374367007575050239, −11.97502490468204974148543904304, −11.54552793670762841363650327821, −11.39452460469157226092504140927, −10.61673933359783053957626960320, −9.540675678002474760589183459020, −9.386041563526135389887346060389, −8.818359049979517687058909589608, −7.75624198653749582752056113611, −7.73364444587185238806396563933, −6.49765882411100238428470306067, −5.83721303326445354613925516785, −5.35349165022672503154391296351, −4.69980943478438159962829993669, −3.38698975598640907057895451127, −3.37928281104705399138313096130, −1.84061157296084221393486606387, 0, 0, 0, 1.84061157296084221393486606387, 3.37928281104705399138313096130, 3.38698975598640907057895451127, 4.69980943478438159962829993669, 5.35349165022672503154391296351, 5.83721303326445354613925516785, 6.49765882411100238428470306067, 7.73364444587185238806396563933, 7.75624198653749582752056113611, 8.818359049979517687058909589608, 9.386041563526135389887346060389, 9.540675678002474760589183459020, 10.61673933359783053957626960320, 11.39452460469157226092504140927, 11.54552793670762841363650327821, 11.97502490468204974148543904304, 12.65337726312374367007575050239, 12.69135701513639340876031820186, 13.89908393504326071701518772159, 14.46065095183698922040384545625, 14.63051300288929024087883648916, 14.93257418061147758179368688434, 15.59945551870535723137625894146, 16.51957543336793069605193661058, 16.86808027554863387250684260478

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