Properties

Label 6-11e3-1.1-c9e3-0-0
Degree $6$
Conductor $1331$
Sign $-1$
Analytic cond. $181.840$
Root an. cond. $2.38020$
Motivic weight $9$
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  − 186·3-s − 312·4-s − 1.82e3·5-s − 7.26e3·7-s − 6.68e3·8-s − 4.95e3·9-s − 4.39e4·11-s + 5.80e4·12-s − 9.32e4·13-s + 3.39e5·15-s − 6.24e4·16-s + 1.86e4·17-s − 1.02e6·19-s + 5.69e5·20-s + 1.35e6·21-s + 1.67e6·23-s + 1.24e6·24-s − 2.04e6·25-s + 8.57e5·27-s + 2.26e6·28-s − 2.69e6·29-s − 4.52e6·31-s + 4.17e6·32-s + 8.16e6·33-s + 1.32e7·35-s + 1.54e6·36-s − 8.82e6·37-s + ⋯
L(s)  = 1  − 1.32·3-s − 0.609·4-s − 1.30·5-s − 1.14·7-s − 0.577·8-s − 0.251·9-s − 0.904·11-s + 0.807·12-s − 0.905·13-s + 1.73·15-s − 0.238·16-s + 0.0542·17-s − 1.80·19-s + 0.795·20-s + 1.51·21-s + 1.24·23-s + 0.765·24-s − 1.04·25-s + 0.310·27-s + 0.696·28-s − 0.707·29-s − 0.880·31-s + 0.703·32-s + 1.19·33-s + 1.49·35-s + 0.153·36-s − 0.773·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s+9/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(1331\)    =    \(11^{3}\)
Sign: $-1$
Analytic conductor: \(181.840\)
Root analytic conductor: \(2.38020\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 1331,\ (\ :9/2, 9/2, 9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) 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$C_1$ \( ( 1 + p^{4} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 39 p^{3} T^{2} + 209 p^{5} T^{3} + 39 p^{12} T^{4} + p^{27} T^{6} \)
3$S_4\times C_2$ \( 1 + 62 p T + 4394 p^{2} T^{2} + 274768 p^{3} T^{3} + 4394 p^{11} T^{4} + 62 p^{19} T^{5} + p^{27} T^{6} \)
5$S_4\times C_2$ \( 1 + 1824 T + 1074708 p T^{2} + 260179306 p^{2} T^{3} + 1074708 p^{10} T^{4} + 1824 p^{18} T^{5} + p^{27} T^{6} \)
7$S_4\times C_2$ \( 1 + 7260 T + 113999049 T^{2} + 72986734040 p T^{3} + 113999049 p^{9} T^{4} + 7260 p^{18} T^{5} + p^{27} T^{6} \)
13$S_4\times C_2$ \( 1 + 93258 T + 30322110999 T^{2} + 1904820864405668 T^{3} + 30322110999 p^{9} T^{4} + 93258 p^{18} T^{5} + p^{27} T^{6} \)
17$S_4\times C_2$ \( 1 - 18678 T + 244701251151 T^{2} + 8819608834790380 T^{3} + 244701251151 p^{9} T^{4} - 18678 p^{18} T^{5} + p^{27} T^{6} \)
19$S_4\times C_2$ \( 1 + 1027356 T + 804683662041 T^{2} + 453396340489692072 T^{3} + 804683662041 p^{9} T^{4} + 1027356 p^{18} T^{5} + p^{27} T^{6} \)
23$S_4\times C_2$ \( 1 - 1674690 T + 1732722881190 T^{2} - 1602963245003190508 T^{3} + 1732722881190 p^{9} T^{4} - 1674690 p^{18} T^{5} + p^{27} T^{6} \)
29$S_4\times C_2$ \( 1 + 2693658 T + 41416876883799 T^{2} + 72041597671730302308 T^{3} + 41416876883799 p^{9} T^{4} + 2693658 p^{18} T^{5} + p^{27} T^{6} \)
31$S_4\times C_2$ \( 1 + 4525302 T + 84482606455422 T^{2} + \)\(23\!\cdots\!20\)\( T^{3} + 84482606455422 p^{9} T^{4} + 4525302 p^{18} T^{5} + p^{27} T^{6} \)
37$S_4\times C_2$ \( 1 + 8820204 T + 323581468812468 T^{2} + \)\(23\!\cdots\!42\)\( T^{3} + 323581468812468 p^{9} T^{4} + 8820204 p^{18} T^{5} + p^{27} T^{6} \)
41$S_4\times C_2$ \( 1 + 9771102 T + 983873066035371 T^{2} + \)\(63\!\cdots\!32\)\( T^{3} + 983873066035371 p^{9} T^{4} + 9771102 p^{18} T^{5} + p^{27} T^{6} \)
43$S_4\times C_2$ \( 1 - 18795744 T + 450693902233077 T^{2} - \)\(63\!\cdots\!96\)\( T^{3} + 450693902233077 p^{9} T^{4} - 18795744 p^{18} T^{5} + p^{27} T^{6} \)
47$S_4\times C_2$ \( 1 + 31155816 T + 1405866049149453 T^{2} + \)\(72\!\cdots\!92\)\( T^{3} + 1405866049149453 p^{9} T^{4} + 31155816 p^{18} T^{5} + p^{27} T^{6} \)
53$S_4\times C_2$ \( 1 - 47500122 T + 7107864087122523 T^{2} - \)\(31\!\cdots\!28\)\( T^{3} + 7107864087122523 p^{9} T^{4} - 47500122 p^{18} T^{5} + p^{27} T^{6} \)
59$S_4\times C_2$ \( 1 - 332138370 T + 61049524588296858 T^{2} - \)\(68\!\cdots\!48\)\( T^{3} + 61049524588296858 p^{9} T^{4} - 332138370 p^{18} T^{5} + p^{27} T^{6} \)
61$S_4\times C_2$ \( 1 + 49031730 T + 18462092029011735 T^{2} + \)\(14\!\cdots\!60\)\( T^{3} + 18462092029011735 p^{9} T^{4} + 49031730 p^{18} T^{5} + p^{27} T^{6} \)
67$S_4\times C_2$ \( 1 - 330560082 T + 88550337598210074 T^{2} - \)\(17\!\cdots\!92\)\( T^{3} + 88550337598210074 p^{9} T^{4} - 330560082 p^{18} T^{5} + p^{27} T^{6} \)
71$S_4\times C_2$ \( 1 + 57835050 T + 107442044673230142 T^{2} + \)\(28\!\cdots\!16\)\( T^{3} + 107442044673230142 p^{9} T^{4} + 57835050 p^{18} T^{5} + p^{27} T^{6} \)
73$S_4\times C_2$ \( 1 + 458816886 T + 179043981976468971 T^{2} + \)\(47\!\cdots\!28\)\( T^{3} + 179043981976468971 p^{9} T^{4} + 458816886 p^{18} T^{5} + p^{27} T^{6} \)
79$S_4\times C_2$ \( 1 + 798908748 T + 300689647984312305 T^{2} + \)\(84\!\cdots\!76\)\( T^{3} + 300689647984312305 p^{9} T^{4} + 798908748 p^{18} T^{5} + p^{27} T^{6} \)
83$S_4\times C_2$ \( 1 - 1239784920 T + 1022523328258118637 T^{2} - \)\(51\!\cdots\!04\)\( T^{3} + 1022523328258118637 p^{9} T^{4} - 1239784920 p^{18} T^{5} + p^{27} T^{6} \)
89$S_4\times C_2$ \( 1 + 699523368 T + 428234191146895896 T^{2} + \)\(90\!\cdots\!94\)\( T^{3} + 428234191146895896 p^{9} T^{4} + 699523368 p^{18} T^{5} + p^{27} T^{6} \)
97$S_4\times C_2$ \( 1 + 2207436012 T + 3730952641114327320 T^{2} + \)\(35\!\cdots\!58\)\( T^{3} + 3730952641114327320 p^{9} T^{4} + 2207436012 p^{18} T^{5} + p^{27} T^{6} \)
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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

−17.36006124867685946634302307651, −16.69422116113071368348011777849, −16.22589999015850201711488560530, −15.99442768024064753403546243900, −15.21054583605370965022929667075, −14.93470858903509320241021772110, −14.55570180570617671039107538143, −13.35261627139822178813434659711, −13.19022460811479344024129075614, −12.68649601141333211969609081861, −11.92342239150368078271068301193, −11.83467452835342509128651440808, −10.94995082471520698560852450286, −10.91930781962024607604700133447, −9.746389424675796263351221684348, −9.534650477708098070183386593283, −8.496898759074231993476957986874, −8.158399499325111867061010824084, −7.13978149903749351023969810175, −6.62401319865353759435682800448, −5.63559385554338713809422228282, −5.35909162917206773108418184488, −4.25399431175674183303402831501, −3.53286253696665252586690407602, −2.45730205558439381856307838203, 0, 0, 0, 2.45730205558439381856307838203, 3.53286253696665252586690407602, 4.25399431175674183303402831501, 5.35909162917206773108418184488, 5.63559385554338713809422228282, 6.62401319865353759435682800448, 7.13978149903749351023969810175, 8.158399499325111867061010824084, 8.496898759074231993476957986874, 9.534650477708098070183386593283, 9.746389424675796263351221684348, 10.91930781962024607604700133447, 10.94995082471520698560852450286, 11.83467452835342509128651440808, 11.92342239150368078271068301193, 12.68649601141333211969609081861, 13.19022460811479344024129075614, 13.35261627139822178813434659711, 14.55570180570617671039107538143, 14.93470858903509320241021772110, 15.21054583605370965022929667075, 15.99442768024064753403546243900, 16.22589999015850201711488560530, 16.69422116113071368348011777849, 17.36006124867685946634302307651

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