Properties

Label 6-3e3-1.1-c35e3-0-0
Degree $6$
Conductor $27$
Sign $1$
Analytic cond. $12614.4$
Root an. cond. $4.82478$
Motivic weight $35$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 8.73e4·2-s − 3.87e8·3-s − 3.14e10·4-s + 2.76e12·5-s + 3.38e13·6-s + 4.88e14·7-s − 2.25e15·8-s + 1.00e17·9-s − 2.41e17·10-s + 3.42e18·11-s + 1.21e19·12-s + 5.00e19·13-s − 4.26e19·14-s − 1.07e21·15-s + 8.19e20·16-s + 2.73e21·17-s − 8.73e21·18-s + 2.87e22·19-s − 8.71e22·20-s − 1.89e23·21-s − 2.99e23·22-s + 1.80e24·23-s + 8.74e23·24-s + 3.76e24·25-s − 4.37e24·26-s − 2.15e25·27-s − 1.53e25·28-s + ⋯
L(s)  = 1  − 0.471·2-s − 1.73·3-s − 0.916·4-s + 1.62·5-s + 0.816·6-s + 0.793·7-s − 0.354·8-s + 2·9-s − 0.764·10-s + 2.04·11-s + 1.58·12-s + 1.60·13-s − 0.373·14-s − 2.81·15-s + 0.694·16-s + 0.800·17-s − 0.942·18-s + 1.20·19-s − 1.48·20-s − 1.37·21-s − 0.962·22-s + 2.67·23-s + 0.613·24-s + 1.29·25-s − 0.756·26-s − 1.92·27-s − 0.726·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(36-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+35/2)^{3} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(27\)    =    \(3^{3}\)
Sign: $1$
Analytic conductor: \(12614.4\)
Root analytic conductor: \(4.82478\)
Motivic weight: \(35\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 27,\ (\ :35/2, 35/2, 35/2),\ 1)\)

Particular Values

\(L(18)\) \(\approx\) \(3.853298082\)
\(L(\frac12)\) \(\approx\) \(3.853298082\)
\(L(\frac{37}{2})\) 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)$
bad3$C_1$ \( ( 1 + p^{17} T )^{3} \)
good2$S_4\times C_2$ \( 1 + 43665 p T + 611070243 p^{6} T^{2} + 256983649773 p^{15} T^{3} + 611070243 p^{41} T^{4} + 43665 p^{71} T^{5} + p^{105} T^{6} \)
5$S_4\times C_2$ \( 1 - 553735247082 p T + \)\(12\!\cdots\!63\)\( p^{5} T^{2} - \)\(24\!\cdots\!64\)\( p^{10} T^{3} + \)\(12\!\cdots\!63\)\( p^{40} T^{4} - 553735247082 p^{71} T^{5} + p^{105} T^{6} \)
7$S_4\times C_2$ \( 1 - 488237848538064 T + \)\(54\!\cdots\!41\)\( p^{2} T^{2} + \)\(35\!\cdots\!96\)\( p^{4} T^{3} + \)\(54\!\cdots\!41\)\( p^{37} T^{4} - 488237848538064 p^{70} T^{5} + p^{105} T^{6} \)
11$S_4\times C_2$ \( 1 - 311486925011293164 p T + \)\(39\!\cdots\!43\)\( p^{3} T^{2} - \)\(43\!\cdots\!60\)\( p^{5} T^{3} + \)\(39\!\cdots\!43\)\( p^{38} T^{4} - 311486925011293164 p^{71} T^{5} + p^{105} T^{6} \)
13$S_4\times C_2$ \( 1 - 50087718431258539506 T + \)\(21\!\cdots\!47\)\( p T^{2} - \)\(35\!\cdots\!96\)\( p^{3} T^{3} + \)\(21\!\cdots\!47\)\( p^{36} T^{4} - 50087718431258539506 p^{70} T^{5} + p^{105} T^{6} \)
17$S_4\times C_2$ \( 1 - \)\(16\!\cdots\!26\)\( p T + \)\(10\!\cdots\!03\)\( p^{2} T^{2} - \)\(12\!\cdots\!12\)\( p^{3} T^{3} + \)\(10\!\cdots\!03\)\( p^{37} T^{4} - \)\(16\!\cdots\!26\)\( p^{71} T^{5} + p^{105} T^{6} \)
19$S_4\times C_2$ \( 1 - \)\(28\!\cdots\!92\)\( T + \)\(13\!\cdots\!93\)\( T^{2} - \)\(11\!\cdots\!84\)\( p T^{3} + \)\(13\!\cdots\!93\)\( p^{35} T^{4} - \)\(28\!\cdots\!92\)\( p^{70} T^{5} + p^{105} T^{6} \)
23$S_4\times C_2$ \( 1 - \)\(18\!\cdots\!08\)\( T + \)\(98\!\cdots\!07\)\( p T^{2} - \)\(33\!\cdots\!28\)\( p^{2} T^{3} + \)\(98\!\cdots\!07\)\( p^{36} T^{4} - \)\(18\!\cdots\!08\)\( p^{70} T^{5} + p^{105} T^{6} \)
29$S_4\times C_2$ \( 1 - \)\(27\!\cdots\!66\)\( p T + \)\(66\!\cdots\!71\)\( p^{2} T^{2} - \)\(92\!\cdots\!68\)\( p^{3} T^{3} + \)\(66\!\cdots\!71\)\( p^{37} T^{4} - \)\(27\!\cdots\!66\)\( p^{71} T^{5} + p^{105} T^{6} \)
31$S_4\times C_2$ \( 1 + \)\(15\!\cdots\!48\)\( T + \)\(10\!\cdots\!63\)\( p T^{2} - \)\(12\!\cdots\!64\)\( p^{2} T^{3} + \)\(10\!\cdots\!63\)\( p^{36} T^{4} + \)\(15\!\cdots\!48\)\( p^{70} T^{5} + p^{105} T^{6} \)
37$S_4\times C_2$ \( 1 + \)\(51\!\cdots\!86\)\( T + \)\(29\!\cdots\!19\)\( T^{2} + \)\(78\!\cdots\!96\)\( T^{3} + \)\(29\!\cdots\!19\)\( p^{35} T^{4} + \)\(51\!\cdots\!86\)\( p^{70} T^{5} + p^{105} T^{6} \)
41$S_4\times C_2$ \( 1 + \)\(20\!\cdots\!58\)\( T + \)\(97\!\cdots\!83\)\( T^{2} + \)\(11\!\cdots\!16\)\( T^{3} + \)\(97\!\cdots\!83\)\( p^{35} T^{4} + \)\(20\!\cdots\!58\)\( p^{70} T^{5} + p^{105} T^{6} \)
43$S_4\times C_2$ \( 1 - \)\(46\!\cdots\!72\)\( T + \)\(36\!\cdots\!57\)\( T^{2} - \)\(11\!\cdots\!00\)\( T^{3} + \)\(36\!\cdots\!57\)\( p^{35} T^{4} - \)\(46\!\cdots\!72\)\( p^{70} T^{5} + p^{105} T^{6} \)
47$S_4\times C_2$ \( 1 - \)\(29\!\cdots\!16\)\( T + \)\(84\!\cdots\!53\)\( T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(84\!\cdots\!53\)\( p^{35} T^{4} - \)\(29\!\cdots\!16\)\( p^{70} T^{5} + p^{105} T^{6} \)
53$S_4\times C_2$ \( 1 + \)\(10\!\cdots\!82\)\( T + \)\(58\!\cdots\!31\)\( T^{2} + \)\(17\!\cdots\!48\)\( T^{3} + \)\(58\!\cdots\!31\)\( p^{35} T^{4} + \)\(10\!\cdots\!82\)\( p^{70} T^{5} + p^{105} T^{6} \)
59$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!48\)\( T + \)\(26\!\cdots\!33\)\( T^{2} - \)\(20\!\cdots\!44\)\( T^{3} + \)\(26\!\cdots\!33\)\( p^{35} T^{4} - \)\(11\!\cdots\!48\)\( p^{70} T^{5} + p^{105} T^{6} \)
61$S_4\times C_2$ \( 1 - \)\(12\!\cdots\!18\)\( T + \)\(44\!\cdots\!99\)\( T^{2} - \)\(42\!\cdots\!64\)\( T^{3} + \)\(44\!\cdots\!99\)\( p^{35} T^{4} - \)\(12\!\cdots\!18\)\( p^{70} T^{5} + p^{105} T^{6} \)
67$S_4\times C_2$ \( 1 - \)\(17\!\cdots\!12\)\( T + \)\(26\!\cdots\!77\)\( T^{2} - \)\(23\!\cdots\!96\)\( T^{3} + \)\(26\!\cdots\!77\)\( p^{35} T^{4} - \)\(17\!\cdots\!12\)\( p^{70} T^{5} + p^{105} T^{6} \)
71$S_4\times C_2$ \( 1 + \)\(27\!\cdots\!24\)\( T + \)\(62\!\cdots\!45\)\( T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(62\!\cdots\!45\)\( p^{35} T^{4} + \)\(27\!\cdots\!24\)\( p^{70} T^{5} + p^{105} T^{6} \)
73$S_4\times C_2$ \( 1 + \)\(58\!\cdots\!22\)\( T + \)\(54\!\cdots\!71\)\( T^{2} + \)\(18\!\cdots\!08\)\( T^{3} + \)\(54\!\cdots\!71\)\( p^{35} T^{4} + \)\(58\!\cdots\!22\)\( p^{70} T^{5} + p^{105} T^{6} \)
79$S_4\times C_2$ \( 1 + \)\(26\!\cdots\!00\)\( T + \)\(99\!\cdots\!97\)\( T^{2} + \)\(14\!\cdots\!00\)\( T^{3} + \)\(99\!\cdots\!97\)\( p^{35} T^{4} + \)\(26\!\cdots\!00\)\( p^{70} T^{5} + p^{105} T^{6} \)
83$S_4\times C_2$ \( 1 - \)\(65\!\cdots\!80\)\( T + \)\(11\!\cdots\!73\)\( T^{2} + \)\(21\!\cdots\!32\)\( T^{3} + \)\(11\!\cdots\!73\)\( p^{35} T^{4} - \)\(65\!\cdots\!80\)\( p^{70} T^{5} + p^{105} T^{6} \)
89$S_4\times C_2$ \( 1 - \)\(11\!\cdots\!22\)\( T + \)\(24\!\cdots\!03\)\( T^{2} - \)\(97\!\cdots\!16\)\( T^{3} + \)\(24\!\cdots\!03\)\( p^{35} T^{4} - \)\(11\!\cdots\!22\)\( p^{70} T^{5} + p^{105} T^{6} \)
97$S_4\times C_2$ \( 1 - \)\(21\!\cdots\!42\)\( T + \)\(24\!\cdots\!67\)\( T^{2} - \)\(17\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!67\)\( p^{35} T^{4} - \)\(21\!\cdots\!42\)\( p^{70} T^{5} + p^{105} T^{6} \)
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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

−15.47095812985426385634500212579, −14.37947788642795473564157206699, −14.19240448294169875785518283962, −13.54633240592906028369853790859, −12.97960243802710428706586304303, −12.19870778338905119894522172707, −11.65320764150379002964282650479, −11.36718097595476200509409895634, −10.30557487035435961757323493970, −10.23224323759285735384010866733, −9.166998023581038281215988487533, −9.048631541195668366788541643019, −8.574298396327967864015148853907, −7.22355413379947825433517940696, −6.47502553658849680481827090440, −6.33721782059236902388913889867, −5.47597542128395986538455733094, −5.23794412592120248455291820731, −4.61535285828647644633611824198, −3.73661901778896157798510472506, −3.10605187498633889435432001298, −1.85550417948922011767130155612, −1.11892760295465658416861557080, −0.934202417721700838595969776949, −0.802255879785529341151295435517, 0.802255879785529341151295435517, 0.934202417721700838595969776949, 1.11892760295465658416861557080, 1.85550417948922011767130155612, 3.10605187498633889435432001298, 3.73661901778896157798510472506, 4.61535285828647644633611824198, 5.23794412592120248455291820731, 5.47597542128395986538455733094, 6.33721782059236902388913889867, 6.47502553658849680481827090440, 7.22355413379947825433517940696, 8.574298396327967864015148853907, 9.048631541195668366788541643019, 9.166998023581038281215988487533, 10.23224323759285735384010866733, 10.30557487035435961757323493970, 11.36718097595476200509409895634, 11.65320764150379002964282650479, 12.19870778338905119894522172707, 12.97960243802710428706586304303, 13.54633240592906028369853790859, 14.19240448294169875785518283962, 14.37947788642795473564157206699, 15.47095812985426385634500212579

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