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Properties

Label 6-5e3-1.1-c23e3-0-0
Degree $6$
Conductor $125$
Sign $-1$
Analytic cond. $4708.01$
Root an. cond. $4.09392$
Motivic weight $23$
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  + 666·2-s − 1.39e5·3-s − 1.71e7·4-s + 1.46e8·5-s − 9.28e7·6-s − 2.43e9·7-s − 1.76e10·8-s − 1.30e11·9-s + 9.75e10·10-s − 1.65e11·11-s + 2.38e12·12-s − 3.55e12·13-s − 1.62e12·14-s − 2.04e13·15-s + 1.44e14·16-s − 4.16e14·17-s − 8.70e13·18-s − 9.75e14·19-s − 2.50e15·20-s + 3.39e14·21-s − 1.10e14·22-s − 4.97e15·23-s + 2.45e15·24-s + 1.43e16·25-s − 2.36e15·26-s + 3.01e16·27-s + 4.16e16·28-s + ⋯
L(s)  = 1  + 0.229·2-s − 0.454·3-s − 2.03·4-s + 1.34·5-s − 0.104·6-s − 0.465·7-s − 0.725·8-s − 1.38·9-s + 0.308·10-s − 0.174·11-s + 0.926·12-s − 0.550·13-s − 0.106·14-s − 0.609·15-s + 2.05·16-s − 2.94·17-s − 0.319·18-s − 1.92·19-s − 2.73·20-s + 0.211·21-s − 0.0401·22-s − 1.08·23-s + 0.329·24-s + 6/5·25-s − 0.126·26-s + 1.04·27-s + 0.948·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 125 ^{s/2} \, \Gamma_{\C}(s+23/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(125\)    =    \(5^{3}\)
Sign: $-1$
Analytic conductor: \(4708.01\)
Root analytic conductor: \(4.09392\)
Motivic weight: \(23\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 125,\ (\ :23/2, 23/2, 23/2),\ -1)\)

Particular Values

\(L(12)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{25}{2})\) 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)$
bad5$C_1$ \( ( 1 - p^{11} T )^{3} \)
good2$S_4\times C_2$ \( 1 - 333 p T + 1096653 p^{4} T^{2} - 5317435 p^{10} T^{3} + 1096653 p^{27} T^{4} - 333 p^{47} T^{5} + p^{69} T^{6} \)
3$S_4\times C_2$ \( 1 + 5164 p^{3} T + 1852942097 p^{4} T^{2} + 455480393480 p^{9} T^{3} + 1852942097 p^{27} T^{4} + 5164 p^{49} T^{5} + p^{69} T^{6} \)
7$S_4\times C_2$ \( 1 + 347526192 p T + 49793351737524051 p^{3} T^{2} + \)\(14\!\cdots\!00\)\( p^{5} T^{3} + 49793351737524051 p^{26} T^{4} + 347526192 p^{47} T^{5} + p^{69} T^{6} \)
11$S_4\times C_2$ \( 1 + 15026002764 p T + \)\(45\!\cdots\!65\)\( p^{2} T^{2} + \)\(10\!\cdots\!80\)\( p^{3} T^{3} + \)\(45\!\cdots\!65\)\( p^{25} T^{4} + 15026002764 p^{47} T^{5} + p^{69} T^{6} \)
13$S_4\times C_2$ \( 1 + 3554970733998 T + \)\(11\!\cdots\!07\)\( T^{2} + \)\(15\!\cdots\!20\)\( p^{2} T^{3} + \)\(11\!\cdots\!07\)\( p^{23} T^{4} + 3554970733998 p^{46} T^{5} + p^{69} T^{6} \)
17$S_4\times C_2$ \( 1 + 416105769269514 T + \)\(79\!\cdots\!83\)\( T^{2} + \)\(66\!\cdots\!40\)\( p T^{3} + \)\(79\!\cdots\!83\)\( p^{23} T^{4} + 416105769269514 p^{46} T^{5} + p^{69} T^{6} \)
19$S_4\times C_2$ \( 1 + 975704043068460 T + \)\(34\!\cdots\!83\)\( p T^{2} + \)\(97\!\cdots\!80\)\( p^{2} T^{3} + \)\(34\!\cdots\!83\)\( p^{24} T^{4} + 975704043068460 p^{46} T^{5} + p^{69} T^{6} \)
23$S_4\times C_2$ \( 1 + 4979666912332368 T + \)\(62\!\cdots\!57\)\( T^{2} + \)\(19\!\cdots\!20\)\( T^{3} + \)\(62\!\cdots\!57\)\( p^{23} T^{4} + 4979666912332368 p^{46} T^{5} + p^{69} T^{6} \)
29$S_4\times C_2$ \( 1 - 95526863867406210 T + \)\(82\!\cdots\!67\)\( T^{2} - \)\(54\!\cdots\!80\)\( T^{3} + \)\(82\!\cdots\!67\)\( p^{23} T^{4} - 95526863867406210 p^{46} T^{5} + p^{69} T^{6} \)
31$S_4\times C_2$ \( 1 - 203997528720796176 T + \)\(65\!\cdots\!65\)\( T^{2} - \)\(81\!\cdots\!20\)\( T^{3} + \)\(65\!\cdots\!65\)\( p^{23} T^{4} - 203997528720796176 p^{46} T^{5} + p^{69} T^{6} \)
37$S_4\times C_2$ \( 1 + 483030177185234454 T + \)\(25\!\cdots\!63\)\( T^{2} + \)\(13\!\cdots\!40\)\( T^{3} + \)\(25\!\cdots\!63\)\( p^{23} T^{4} + 483030177185234454 p^{46} T^{5} + p^{69} T^{6} \)
41$S_4\times C_2$ \( 1 + 4358124871734570834 T + \)\(28\!\cdots\!15\)\( T^{2} + \)\(10\!\cdots\!80\)\( T^{3} + \)\(28\!\cdots\!15\)\( p^{23} T^{4} + 4358124871734570834 p^{46} T^{5} + p^{69} T^{6} \)
43$S_4\times C_2$ \( 1 + 5324153686845536508 T + \)\(87\!\cdots\!57\)\( T^{2} + \)\(27\!\cdots\!00\)\( T^{3} + \)\(87\!\cdots\!57\)\( p^{23} T^{4} + 5324153686845536508 p^{46} T^{5} + p^{69} T^{6} \)
47$S_4\times C_2$ \( 1 - 12258262539323374776 T + \)\(10\!\cdots\!53\)\( T^{2} - \)\(16\!\cdots\!80\)\( T^{3} + \)\(10\!\cdots\!53\)\( p^{23} T^{4} - 12258262539323374776 p^{46} T^{5} + p^{69} T^{6} \)
53$S_4\times C_2$ \( 1 - 71640983942760085722 T + \)\(10\!\cdots\!07\)\( T^{2} - \)\(61\!\cdots\!60\)\( T^{3} + \)\(10\!\cdots\!07\)\( p^{23} T^{4} - 71640983942760085722 p^{46} T^{5} + p^{69} T^{6} \)
59$S_4\times C_2$ \( 1 + \)\(42\!\cdots\!80\)\( T + \)\(20\!\cdots\!37\)\( T^{2} + \)\(45\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!37\)\( p^{23} T^{4} + \)\(42\!\cdots\!80\)\( p^{46} T^{5} + p^{69} T^{6} \)
61$S_4\times C_2$ \( 1 + \)\(58\!\cdots\!54\)\( T + \)\(44\!\cdots\!15\)\( T^{2} + \)\(13\!\cdots\!80\)\( T^{3} + \)\(44\!\cdots\!15\)\( p^{23} T^{4} + \)\(58\!\cdots\!54\)\( p^{46} T^{5} + p^{69} T^{6} \)
67$S_4\times C_2$ \( 1 + \)\(13\!\cdots\!64\)\( T + \)\(31\!\cdots\!33\)\( T^{2} + \)\(26\!\cdots\!80\)\( T^{3} + \)\(31\!\cdots\!33\)\( p^{23} T^{4} + \)\(13\!\cdots\!64\)\( p^{46} T^{5} + p^{69} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(40\!\cdots\!64\)\( T + \)\(76\!\cdots\!65\)\( T^{2} + \)\(50\!\cdots\!80\)\( T^{3} + \)\(76\!\cdots\!65\)\( p^{23} T^{4} + \)\(40\!\cdots\!64\)\( p^{46} T^{5} + p^{69} T^{6} \)
73$S_4\times C_2$ \( 1 - \)\(32\!\cdots\!82\)\( T + \)\(22\!\cdots\!07\)\( T^{2} - \)\(44\!\cdots\!80\)\( T^{3} + \)\(22\!\cdots\!07\)\( p^{23} T^{4} - \)\(32\!\cdots\!82\)\( p^{46} T^{5} + p^{69} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!40\)\( T + \)\(15\!\cdots\!17\)\( T^{2} + \)\(90\!\cdots\!20\)\( T^{3} + \)\(15\!\cdots\!17\)\( p^{23} T^{4} + \)\(10\!\cdots\!40\)\( p^{46} T^{5} + p^{69} T^{6} \)
83$S_4\times C_2$ \( 1 + \)\(14\!\cdots\!88\)\( T + \)\(33\!\cdots\!57\)\( T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(33\!\cdots\!57\)\( p^{23} T^{4} + \)\(14\!\cdots\!88\)\( p^{46} T^{5} + p^{69} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(55\!\cdots\!30\)\( T + \)\(29\!\cdots\!07\)\( T^{2} - \)\(79\!\cdots\!40\)\( T^{3} + \)\(29\!\cdots\!07\)\( p^{23} T^{4} - \)\(55\!\cdots\!30\)\( p^{46} T^{5} + p^{69} T^{6} \)
97$S_4\times C_2$ \( 1 + \)\(23\!\cdots\!74\)\( T + \)\(32\!\cdots\!03\)\( T^{2} + \)\(27\!\cdots\!20\)\( T^{3} + \)\(32\!\cdots\!03\)\( p^{23} T^{4} + \)\(23\!\cdots\!74\)\( p^{46} T^{5} + p^{69} T^{6} \)
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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

−17.13570905014583392486141747838, −16.19559586326048970169002729758, −15.10939920669161262493231653309, −15.07641405689013413350955401138, −13.83337364403607488605640018873, −13.74090675738545802532976829209, −13.71318615604401779525901392276, −12.83849895479429370792483608758, −12.39037570666784245071830470186, −11.64097412499975473174530099918, −10.67308614501053613510628283000, −10.31434744629029332781650606138, −9.630701723119307608570316232655, −8.913773679543700067436981429134, −8.662072177564089487383224736030, −8.338295243056947317669193501559, −6.51536500727386182175920052570, −6.44715522863965432700416689566, −5.76440815429138473858841946767, −4.76694092180680855717465167691, −4.73692437868497777512787410532, −4.01309005141254904445377829160, −2.65179209199316653001894435455, −2.53985022148423492135862621662, −1.43037514122873215593986529295, 0, 0, 0, 1.43037514122873215593986529295, 2.53985022148423492135862621662, 2.65179209199316653001894435455, 4.01309005141254904445377829160, 4.73692437868497777512787410532, 4.76694092180680855717465167691, 5.76440815429138473858841946767, 6.44715522863965432700416689566, 6.51536500727386182175920052570, 8.338295243056947317669193501559, 8.662072177564089487383224736030, 8.913773679543700067436981429134, 9.630701723119307608570316232655, 10.31434744629029332781650606138, 10.67308614501053613510628283000, 11.64097412499975473174530099918, 12.39037570666784245071830470186, 12.83849895479429370792483608758, 13.71318615604401779525901392276, 13.74090675738545802532976829209, 13.83337364403607488605640018873, 15.07641405689013413350955401138, 15.10939920669161262493231653309, 16.19559586326048970169002729758, 17.13570905014583392486141747838

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