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Properties

Label 6-2e6-1.1-c35e3-0-0
Degree $6$
Conductor $64$
Sign $-1$
Analytic cond. $29900.8$
Root an. cond. $5.57118$
Motivic weight $35$
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  + 5.09e7·3-s + 2.80e8·5-s − 5.54e12·7-s − 7.58e16·9-s − 4.20e17·11-s − 7.37e18·13-s + 1.42e16·15-s + 6.16e20·17-s − 3.52e21·19-s − 2.82e20·21-s − 9.72e22·23-s − 5.06e24·25-s − 9.35e24·27-s − 7.21e25·29-s − 3.05e26·31-s − 2.14e25·33-s − 1.55e21·35-s − 1.04e28·37-s − 3.75e26·39-s − 2.86e28·41-s − 2.06e28·43-s − 2.12e25·45-s + 4.31e29·47-s + 3.29e28·49-s + 3.14e28·51-s + 2.09e30·53-s − 1.18e26·55-s + ⋯
L(s)  = 1  + 0.227·3-s + 0.000164·5-s − 0.00901·7-s − 1.51·9-s − 0.250·11-s − 0.236·13-s + 3.74e−5·15-s + 0.180·17-s − 0.147·19-s − 0.00205·21-s − 0.143·23-s − 1.74·25-s − 0.836·27-s − 1.84·29-s − 2.43·31-s − 0.0570·33-s − 1.48e − 6·35-s − 3.76·37-s − 0.0537·39-s − 1.70·41-s − 0.536·43-s − 0.000249·45-s + 2.36·47-s + 0.0869·49-s + 0.0411·51-s + 1.40·53-s − 4.12e − 5·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+35/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $-1$
Analytic conductor: \(29900.8\)
Root analytic conductor: \(5.57118\)
Motivic weight: \(35\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 64,\ (\ :35/2, 35/2, 35/2),\ -1)\)

Particular Values

\(L(18)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{37}{2})\) 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 \( 1 \)
good3$S_4\times C_2$ \( 1 - 16969628 p T + 322637950381115 p^{5} T^{2} + 2834011306570298888 p^{12} T^{3} + 322637950381115 p^{40} T^{4} - 16969628 p^{71} T^{5} + p^{105} T^{6} \)
5$S_4\times C_2$ \( 1 - 56144178 p T + \)\(16\!\cdots\!83\)\( p^{5} T^{2} + \)\(17\!\cdots\!84\)\( p^{10} T^{3} + \)\(16\!\cdots\!83\)\( p^{40} T^{4} - 56144178 p^{71} T^{5} + p^{105} T^{6} \)
7$S_4\times C_2$ \( 1 + 5549296289016 T - \)\(67\!\cdots\!83\)\( p^{2} T^{2} - \)\(11\!\cdots\!60\)\( p^{4} T^{3} - \)\(67\!\cdots\!83\)\( p^{37} T^{4} + 5549296289016 p^{70} T^{5} + p^{105} T^{6} \)
11$S_4\times C_2$ \( 1 + 420463958869151460 T + \)\(29\!\cdots\!93\)\( p^{2} T^{2} - \)\(83\!\cdots\!80\)\( p^{4} T^{3} + \)\(29\!\cdots\!93\)\( p^{37} T^{4} + 420463958869151460 p^{70} T^{5} + p^{105} T^{6} \)
13$S_4\times C_2$ \( 1 + 7370701315414736574 T + \)\(97\!\cdots\!55\)\( p T^{2} + \)\(49\!\cdots\!36\)\( p^{3} T^{3} + \)\(97\!\cdots\!55\)\( p^{36} T^{4} + 7370701315414736574 p^{70} T^{5} + p^{105} T^{6} \)
17$S_4\times C_2$ \( 1 - 36282486732283653174 p T + \)\(48\!\cdots\!15\)\( p^{2} T^{2} - \)\(13\!\cdots\!72\)\( p^{3} T^{3} + \)\(48\!\cdots\!15\)\( p^{37} T^{4} - 36282486732283653174 p^{71} T^{5} + p^{105} T^{6} \)
19$S_4\times C_2$ \( 1 + \)\(35\!\cdots\!48\)\( T + \)\(89\!\cdots\!65\)\( T^{2} + \)\(29\!\cdots\!00\)\( p^{2} T^{3} + \)\(89\!\cdots\!65\)\( p^{35} T^{4} + \)\(35\!\cdots\!48\)\( p^{70} T^{5} + p^{105} T^{6} \)
23$S_4\times C_2$ \( 1 + \)\(97\!\cdots\!72\)\( T + \)\(35\!\cdots\!39\)\( p T^{2} - \)\(17\!\cdots\!80\)\( p^{2} T^{3} + \)\(35\!\cdots\!39\)\( p^{36} T^{4} + \)\(97\!\cdots\!72\)\( p^{70} T^{5} + p^{105} T^{6} \)
29$S_4\times C_2$ \( 1 + \)\(24\!\cdots\!62\)\( p T + \)\(72\!\cdots\!15\)\( p^{2} T^{2} + \)\(93\!\cdots\!00\)\( p^{3} T^{3} + \)\(72\!\cdots\!15\)\( p^{37} T^{4} + \)\(24\!\cdots\!62\)\( p^{71} T^{5} + p^{105} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(30\!\cdots\!32\)\( T + \)\(22\!\cdots\!31\)\( p T^{2} + \)\(99\!\cdots\!68\)\( p^{2} T^{3} + \)\(22\!\cdots\!31\)\( p^{36} T^{4} + \)\(30\!\cdots\!32\)\( p^{70} T^{5} + p^{105} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!86\)\( T + \)\(59\!\cdots\!23\)\( T^{2} + \)\(20\!\cdots\!80\)\( T^{3} + \)\(59\!\cdots\!23\)\( p^{35} T^{4} + \)\(10\!\cdots\!86\)\( p^{70} T^{5} + p^{105} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(28\!\cdots\!02\)\( T + \)\(31\!\cdots\!71\)\( T^{2} + \)\(13\!\cdots\!08\)\( T^{3} + \)\(31\!\cdots\!71\)\( p^{35} T^{4} + \)\(28\!\cdots\!02\)\( p^{70} T^{5} + p^{105} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(20\!\cdots\!00\)\( T + \)\(15\!\cdots\!21\)\( T^{2} + \)\(95\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!21\)\( p^{35} T^{4} + \)\(20\!\cdots\!00\)\( p^{70} T^{5} + p^{105} T^{6} \)
47$S_4\times C_2$ \( 1 - \)\(43\!\cdots\!28\)\( T + \)\(93\!\cdots\!45\)\( T^{2} - \)\(14\!\cdots\!56\)\( T^{3} + \)\(93\!\cdots\!45\)\( p^{35} T^{4} - \)\(43\!\cdots\!28\)\( p^{70} T^{5} + p^{105} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(20\!\cdots\!42\)\( T + \)\(66\!\cdots\!87\)\( T^{2} - \)\(80\!\cdots\!60\)\( T^{3} + \)\(66\!\cdots\!87\)\( p^{35} T^{4} - \)\(20\!\cdots\!42\)\( p^{70} T^{5} + p^{105} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(21\!\cdots\!76\)\( T + \)\(40\!\cdots\!89\)\( T^{2} + \)\(43\!\cdots\!36\)\( T^{3} + \)\(40\!\cdots\!89\)\( p^{35} T^{4} + \)\(21\!\cdots\!76\)\( p^{70} T^{5} + p^{105} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(46\!\cdots\!54\)\( T + \)\(14\!\cdots\!75\)\( T^{2} + \)\(30\!\cdots\!40\)\( T^{3} + \)\(14\!\cdots\!75\)\( p^{35} T^{4} + \)\(46\!\cdots\!54\)\( p^{70} T^{5} + p^{105} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(75\!\cdots\!84\)\( T + \)\(14\!\cdots\!13\)\( T^{2} - \)\(46\!\cdots\!80\)\( T^{3} + \)\(14\!\cdots\!13\)\( p^{35} T^{4} - \)\(75\!\cdots\!84\)\( p^{70} T^{5} + p^{105} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(59\!\cdots\!44\)\( T + \)\(29\!\cdots\!65\)\( T^{2} - \)\(80\!\cdots\!80\)\( T^{3} + \)\(29\!\cdots\!65\)\( p^{35} T^{4} - \)\(59\!\cdots\!44\)\( p^{70} T^{5} + p^{105} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(84\!\cdots\!66\)\( T + \)\(49\!\cdots\!35\)\( T^{2} - \)\(19\!\cdots\!48\)\( T^{3} + \)\(49\!\cdots\!35\)\( p^{35} T^{4} - \)\(84\!\cdots\!66\)\( p^{70} T^{5} + p^{105} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!44\)\( T + \)\(80\!\cdots\!09\)\( T^{2} + \)\(76\!\cdots\!04\)\( T^{3} + \)\(80\!\cdots\!09\)\( p^{35} T^{4} + \)\(15\!\cdots\!44\)\( p^{70} T^{5} + p^{105} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(89\!\cdots\!12\)\( T + \)\(61\!\cdots\!77\)\( T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(61\!\cdots\!77\)\( p^{35} T^{4} + \)\(89\!\cdots\!12\)\( p^{70} T^{5} + p^{105} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(16\!\cdots\!98\)\( T + \)\(16\!\cdots\!15\)\( T^{2} - \)\(21\!\cdots\!00\)\( T^{3} + \)\(16\!\cdots\!15\)\( p^{35} T^{4} - \)\(16\!\cdots\!98\)\( p^{70} T^{5} + p^{105} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(83\!\cdots\!14\)\( T + \)\(11\!\cdots\!03\)\( T^{2} - \)\(57\!\cdots\!40\)\( T^{3} + \)\(11\!\cdots\!03\)\( p^{35} T^{4} - \)\(83\!\cdots\!14\)\( p^{70} T^{5} + p^{105} T^{6} \)
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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

−15.15775775512923627644159452633, −14.28315083837135092532429722256, −13.78046311136324036726035846459, −13.73120124789628104874772643216, −12.78844986063243266000246045441, −12.01450747588049854028099336865, −12.01305186546829715990166093910, −10.92751022972482810258196771346, −10.89323781289668567926746660280, −10.08489857034891926239279373635, −9.168254892874601745910391238951, −9.127427940444475486101980618772, −8.381518577231603154586730584182, −7.78130390252384120519825743408, −7.29222897457616762554052815458, −6.69044204675563188791682193526, −5.69946478844270906508958494548, −5.50821052610249223650900278030, −5.17626884591569885794689381459, −3.88303508670677331735478864143, −3.70063862090065451799685174861, −3.15638181419415468202331382644, −2.30923951838911526524243083161, −1.88037850481774459175859408422, −1.48409086764138366510980314082, 0, 0, 0, 1.48409086764138366510980314082, 1.88037850481774459175859408422, 2.30923951838911526524243083161, 3.15638181419415468202331382644, 3.70063862090065451799685174861, 3.88303508670677331735478864143, 5.17626884591569885794689381459, 5.50821052610249223650900278030, 5.69946478844270906508958494548, 6.69044204675563188791682193526, 7.29222897457616762554052815458, 7.78130390252384120519825743408, 8.381518577231603154586730584182, 9.127427940444475486101980618772, 9.168254892874601745910391238951, 10.08489857034891926239279373635, 10.89323781289668567926746660280, 10.92751022972482810258196771346, 12.01305186546829715990166093910, 12.01450747588049854028099336865, 12.78844986063243266000246045441, 13.73120124789628104874772643216, 13.78046311136324036726035846459, 14.28315083837135092532429722256, 15.15775775512923627644159452633

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