Properties

Degree $6$
Conductor $64$
Sign $1$
Motivic weight $33$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.24e7·3-s − 5.38e10·5-s + 4.54e12·7-s − 1.04e15·9-s + 2.27e17·11-s + 2.72e17·13-s − 4.98e18·15-s + 9.30e19·17-s + 1.32e21·19-s + 4.20e20·21-s + 1.16e22·23-s − 4.40e22·25-s − 1.43e23·27-s + 2.89e24·29-s + 1.58e25·31-s + 2.10e25·33-s − 2.44e23·35-s + 3.47e25·37-s + 2.52e25·39-s − 6.73e26·41-s − 1.55e27·43-s + 5.63e25·45-s − 7.22e27·47-s − 1.39e28·49-s + 8.60e27·51-s + 5.40e28·53-s − 1.22e28·55-s + ⋯
L(s)  = 1  + 1.24·3-s − 0.157·5-s + 0.0516·7-s − 0.187·9-s + 1.49·11-s + 0.113·13-s − 0.195·15-s + 0.463·17-s + 1.05·19-s + 0.0640·21-s + 0.397·23-s − 0.378·25-s − 0.345·27-s + 2.15·29-s + 3.90·31-s + 1.85·33-s − 0.00815·35-s + 0.463·37-s + 0.141·39-s − 1.65·41-s − 1.73·43-s + 0.0296·45-s − 1.85·47-s − 1.80·49-s + 0.575·51-s + 1.91·53-s − 0.235·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+33/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(64\)    =    \(2^{6}\)
Sign: $1$
Motivic weight: \(33\)
Character: induced by $\chi_{4} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 64,\ (\ :33/2, 33/2, 33/2),\ 1)\)

Particular Values

\(L(17)\) \(\approx\) \(7.111810932\)
\(L(\frac12)\) \(\approx\) \(7.111810932\)
\(L(\frac{35}{2})\) 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$S_4\times C_2$ \( 1 - 30830596 p T + 13168414343905 p^{6} T^{2} - 527785882937478232 p^{13} T^{3} + 13168414343905 p^{39} T^{4} - 30830596 p^{67} T^{5} + p^{99} T^{6} \)
5$S_4\times C_2$ \( 1 + 53880683886 T + \)\(37\!\cdots\!83\)\( p^{3} T^{2} - \)\(11\!\cdots\!32\)\( p^{8} T^{3} + \)\(37\!\cdots\!83\)\( p^{36} T^{4} + 53880683886 p^{66} T^{5} + p^{99} T^{6} \)
7$S_4\times C_2$ \( 1 - 648715702056 p T + \)\(28\!\cdots\!53\)\( p^{2} T^{2} - \)\(33\!\cdots\!20\)\( p^{6} T^{3} + \)\(28\!\cdots\!53\)\( p^{35} T^{4} - 648715702056 p^{67} T^{5} + p^{99} T^{6} \)
11$S_4\times C_2$ \( 1 - 20692514300222700 p T + \)\(16\!\cdots\!33\)\( p^{2} T^{2} + \)\(53\!\cdots\!00\)\( p^{3} T^{3} + \)\(16\!\cdots\!33\)\( p^{35} T^{4} - 20692514300222700 p^{67} T^{5} + p^{99} T^{6} \)
13$S_4\times C_2$ \( 1 - 272970442217358762 T + \)\(10\!\cdots\!15\)\( p T^{2} - \)\(12\!\cdots\!32\)\( p^{3} T^{3} + \)\(10\!\cdots\!15\)\( p^{34} T^{4} - 272970442217358762 p^{66} T^{5} + p^{99} T^{6} \)
17$S_4\times C_2$ \( 1 - 93037188311816716854 T + \)\(10\!\cdots\!75\)\( T^{2} - \)\(44\!\cdots\!56\)\( p T^{3} + \)\(10\!\cdots\!75\)\( p^{33} T^{4} - 93037188311816716854 p^{66} T^{5} + p^{99} T^{6} \)
19$S_4\times C_2$ \( 1 - 3670049324584713948 p^{2} T + \)\(20\!\cdots\!95\)\( p T^{2} - \)\(97\!\cdots\!80\)\( p^{2} T^{3} + \)\(20\!\cdots\!95\)\( p^{34} T^{4} - 3670049324584713948 p^{68} T^{5} + p^{99} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!96\)\( T + \)\(81\!\cdots\!51\)\( p T^{2} - \)\(28\!\cdots\!60\)\( p^{2} T^{3} + \)\(81\!\cdots\!51\)\( p^{34} T^{4} - \)\(11\!\cdots\!96\)\( p^{66} T^{5} + p^{99} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(28\!\cdots\!78\)\( T + \)\(15\!\cdots\!55\)\( p T^{2} - \)\(56\!\cdots\!60\)\( p^{2} T^{3} + \)\(15\!\cdots\!55\)\( p^{34} T^{4} - \)\(28\!\cdots\!78\)\( p^{66} T^{5} + p^{99} T^{6} \)
31$S_4\times C_2$ \( 1 - \)\(51\!\cdots\!88\)\( p T + \)\(13\!\cdots\!41\)\( p^{2} T^{2} - \)\(22\!\cdots\!92\)\( p^{3} T^{3} + \)\(13\!\cdots\!41\)\( p^{35} T^{4} - \)\(51\!\cdots\!88\)\( p^{67} T^{5} + p^{99} T^{6} \)
37$S_4\times C_2$ \( 1 - \)\(34\!\cdots\!42\)\( T + \)\(58\!\cdots\!87\)\( T^{2} - \)\(50\!\cdots\!80\)\( T^{3} + \)\(58\!\cdots\!87\)\( p^{33} T^{4} - \)\(34\!\cdots\!42\)\( p^{66} T^{5} + p^{99} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(67\!\cdots\!62\)\( T + \)\(59\!\cdots\!11\)\( T^{2} + \)\(21\!\cdots\!68\)\( T^{3} + \)\(59\!\cdots\!11\)\( p^{33} T^{4} + \)\(67\!\cdots\!62\)\( p^{66} T^{5} + p^{99} T^{6} \)
43$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!20\)\( T + \)\(22\!\cdots\!29\)\( T^{2} + \)\(45\!\cdots\!40\)\( p T^{3} + \)\(22\!\cdots\!29\)\( p^{33} T^{4} + \)\(15\!\cdots\!20\)\( p^{66} T^{5} + p^{99} T^{6} \)
47$S_4\times C_2$ \( 1 + \)\(72\!\cdots\!76\)\( T + \)\(60\!\cdots\!05\)\( T^{2} + \)\(22\!\cdots\!28\)\( T^{3} + \)\(60\!\cdots\!05\)\( p^{33} T^{4} + \)\(72\!\cdots\!76\)\( p^{66} T^{5} + p^{99} T^{6} \)
53$S_4\times C_2$ \( 1 - \)\(54\!\cdots\!34\)\( T + \)\(28\!\cdots\!83\)\( T^{2} - \)\(80\!\cdots\!40\)\( T^{3} + \)\(28\!\cdots\!83\)\( p^{33} T^{4} - \)\(54\!\cdots\!34\)\( p^{66} T^{5} + p^{99} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(20\!\cdots\!76\)\( T + \)\(56\!\cdots\!29\)\( T^{2} - \)\(65\!\cdots\!96\)\( T^{3} + \)\(56\!\cdots\!29\)\( p^{33} T^{4} - \)\(20\!\cdots\!76\)\( p^{66} T^{5} + p^{99} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(59\!\cdots\!66\)\( T + \)\(26\!\cdots\!95\)\( T^{2} - \)\(28\!\cdots\!40\)\( T^{3} + \)\(26\!\cdots\!95\)\( p^{33} T^{4} - \)\(59\!\cdots\!66\)\( p^{66} T^{5} + p^{99} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(29\!\cdots\!28\)\( T + \)\(81\!\cdots\!37\)\( T^{2} + \)\(11\!\cdots\!00\)\( T^{3} + \)\(81\!\cdots\!37\)\( p^{33} T^{4} + \)\(29\!\cdots\!28\)\( p^{66} T^{5} + p^{99} T^{6} \)
71$S_4\times C_2$ \( 1 - \)\(16\!\cdots\!44\)\( T + \)\(30\!\cdots\!45\)\( T^{2} - \)\(32\!\cdots\!60\)\( T^{3} + \)\(30\!\cdots\!45\)\( p^{33} T^{4} - \)\(16\!\cdots\!44\)\( p^{66} T^{5} + p^{99} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(13\!\cdots\!02\)\( T + \)\(36\!\cdots\!75\)\( T^{2} - \)\(19\!\cdots\!84\)\( T^{3} + \)\(36\!\cdots\!75\)\( p^{33} T^{4} - \)\(13\!\cdots\!02\)\( p^{66} T^{5} + p^{99} T^{6} \)
79$S_4\times C_2$ \( 1 - \)\(43\!\cdots\!04\)\( T + \)\(15\!\cdots\!89\)\( T^{2} - \)\(35\!\cdots\!44\)\( T^{3} + \)\(15\!\cdots\!89\)\( p^{33} T^{4} - \)\(43\!\cdots\!04\)\( p^{66} T^{5} + p^{99} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(71\!\cdots\!64\)\( T + \)\(66\!\cdots\!93\)\( T^{2} + \)\(26\!\cdots\!20\)\( T^{3} + \)\(66\!\cdots\!93\)\( p^{33} T^{4} + \)\(71\!\cdots\!64\)\( p^{66} T^{5} + p^{99} T^{6} \)
89$S_4\times C_2$ \( 1 + \)\(19\!\cdots\!98\)\( T + \)\(63\!\cdots\!75\)\( T^{2} + \)\(76\!\cdots\!20\)\( T^{3} + \)\(63\!\cdots\!75\)\( p^{33} T^{4} + \)\(19\!\cdots\!98\)\( p^{66} T^{5} + p^{99} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(56\!\cdots\!22\)\( T - \)\(50\!\cdots\!73\)\( T^{2} + \)\(35\!\cdots\!00\)\( T^{3} - \)\(50\!\cdots\!73\)\( p^{33} T^{4} - \)\(56\!\cdots\!22\)\( p^{66} T^{5} + p^{99} T^{6} \)
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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

−14.46332418152513106338421879764, −13.93535191001986867513127557801, −13.72704328342322776545814465675, −13.17541531217090719511557393482, −11.99025560246433380318376728905, −11.73590723627168521649275674842, −11.69519008438609794438957379778, −10.37787385145013433546168706059, −9.890616235970197654756947968842, −9.567785108082663772776959964757, −8.549824988887918785174758742834, −8.362685900156924478337277592799, −8.198090569384290375168761949202, −7.09976330916847255445578417588, −6.38607836330697214818893296773, −6.34431902832542930260175433855, −4.90104549828883922952571876861, −4.86824427702834250115861835213, −3.86237457259214702047208929759, −3.26949025307006799163845041189, −2.92070769047758086335416690277, −2.48515890102930602818381678793, −1.36732755296961624989494891612, −1.21308022921594800971515460596, −0.50970554940105003785325212622, 0.50970554940105003785325212622, 1.21308022921594800971515460596, 1.36732755296961624989494891612, 2.48515890102930602818381678793, 2.92070769047758086335416690277, 3.26949025307006799163845041189, 3.86237457259214702047208929759, 4.86824427702834250115861835213, 4.90104549828883922952571876861, 6.34431902832542930260175433855, 6.38607836330697214818893296773, 7.09976330916847255445578417588, 8.198090569384290375168761949202, 8.362685900156924478337277592799, 8.549824988887918785174758742834, 9.567785108082663772776959964757, 9.890616235970197654756947968842, 10.37787385145013433546168706059, 11.69519008438609794438957379778, 11.73590723627168521649275674842, 11.99025560246433380318376728905, 13.17541531217090719511557393482, 13.72704328342322776545814465675, 13.93535191001986867513127557801, 14.46332418152513106338421879764

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