Properties

Label 6-165e3-1.1-c5e3-0-2
Degree $6$
Conductor $4492125$
Sign $-1$
Analytic cond. $18532.4$
Root an. cond. $5.14425$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 27·3-s − 67·4-s + 75·5-s − 54·6-s − 68·7-s − 166·8-s + 486·9-s + 150·10-s + 363·11-s + 1.80e3·12-s + 290·13-s − 136·14-s − 2.02e3·15-s + 2.37e3·16-s + 434·17-s + 972·18-s − 2.85e3·19-s − 5.02e3·20-s + 1.83e3·21-s + 726·22-s − 640·23-s + 4.48e3·24-s + 3.75e3·25-s + 580·26-s − 7.29e3·27-s + 4.55e3·28-s + ⋯
L(s)  = 1  + 0.353·2-s − 1.73·3-s − 2.09·4-s + 1.34·5-s − 0.612·6-s − 0.524·7-s − 0.917·8-s + 2·9-s + 0.474·10-s + 0.904·11-s + 3.62·12-s + 0.475·13-s − 0.185·14-s − 2.32·15-s + 2.31·16-s + 0.364·17-s + 0.707·18-s − 1.81·19-s − 2.80·20-s + 0.908·21-s + 0.319·22-s − 0.252·23-s + 1.58·24-s + 6/5·25-s + 0.168·26-s − 1.92·27-s + 1.09·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4492125 ^{s/2} \, \Gamma_{\C}(s+5/2)^{3} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(4492125\)    =    \(3^{3} \cdot 5^{3} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(18532.4\)
Root analytic conductor: \(5.14425\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 4492125,\ (\ :5/2, 5/2, 5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 + p^{2} T )^{3} \)
5$C_1$ \( ( 1 - p^{2} T )^{3} \)
11$C_1$ \( ( 1 - p^{2} T )^{3} \)
good2$S_4\times C_2$ \( 1 - p T + 71 T^{2} - 55 p T^{3} + 71 p^{5} T^{4} - p^{11} T^{5} + p^{15} T^{6} \)
7$S_4\times C_2$ \( 1 + 68 T + 34869 T^{2} + 2533560 T^{3} + 34869 p^{5} T^{4} + 68 p^{10} T^{5} + p^{15} T^{6} \)
13$S_4\times C_2$ \( 1 - 290 T + 658819 T^{2} - 83286348 T^{3} + 658819 p^{5} T^{4} - 290 p^{10} T^{5} + p^{15} T^{6} \)
17$S_4\times C_2$ \( 1 - 434 T + 58991 T^{2} + 1314616612 T^{3} + 58991 p^{5} T^{4} - 434 p^{10} T^{5} + p^{15} T^{6} \)
19$S_4\times C_2$ \( 1 + 2856 T + 8753113 T^{2} + 14281181168 T^{3} + 8753113 p^{5} T^{4} + 2856 p^{10} T^{5} + p^{15} T^{6} \)
23$S_4\times C_2$ \( 1 + 640 T + 6700261 T^{2} - 7538830592 T^{3} + 6700261 p^{5} T^{4} + 640 p^{10} T^{5} + p^{15} T^{6} \)
29$S_4\times C_2$ \( 1 + 4538 T + 35092643 T^{2} + 141745639868 T^{3} + 35092643 p^{5} T^{4} + 4538 p^{10} T^{5} + p^{15} T^{6} \)
31$S_4\times C_2$ \( 1 + 14968 T + 160071261 T^{2} + 978687787280 T^{3} + 160071261 p^{5} T^{4} + 14968 p^{10} T^{5} + p^{15} T^{6} \)
37$S_4\times C_2$ \( 1 + 6190 T + 113089771 T^{2} + 886491849396 T^{3} + 113089771 p^{5} T^{4} + 6190 p^{10} T^{5} + p^{15} T^{6} \)
41$S_4\times C_2$ \( 1 + 8926 T + 272145047 T^{2} + 2187570068644 T^{3} + 272145047 p^{5} T^{4} + 8926 p^{10} T^{5} + p^{15} T^{6} \)
43$S_4\times C_2$ \( 1 + 33592 T + 693464257 T^{2} + 9837617686992 T^{3} + 693464257 p^{5} T^{4} + 33592 p^{10} T^{5} + p^{15} T^{6} \)
47$S_4\times C_2$ \( 1 + 24640 T + 788101261 T^{2} + 10622124840832 T^{3} + 788101261 p^{5} T^{4} + 24640 p^{10} T^{5} + p^{15} T^{6} \)
53$S_4\times C_2$ \( 1 + 22934 T + 1099334651 T^{2} + 14788031800868 T^{3} + 1099334651 p^{5} T^{4} + 22934 p^{10} T^{5} + p^{15} T^{6} \)
59$S_4\times C_2$ \( 1 + 13756 T - 129165359 T^{2} - 1129007325848 T^{3} - 129165359 p^{5} T^{4} + 13756 p^{10} T^{5} + p^{15} T^{6} \)
61$S_4\times C_2$ \( 1 - 24602 T + 2638007155 T^{2} - 41514451599900 T^{3} + 2638007155 p^{5} T^{4} - 24602 p^{10} T^{5} + p^{15} T^{6} \)
67$S_4\times C_2$ \( 1 - 16868 T + 3843672649 T^{2} - 43721067889560 T^{3} + 3843672649 p^{5} T^{4} - 16868 p^{10} T^{5} + p^{15} T^{6} \)
71$S_4\times C_2$ \( 1 - 4856 T + 1951490165 T^{2} + 16632390591088 T^{3} + 1951490165 p^{5} T^{4} - 4856 p^{10} T^{5} + p^{15} T^{6} \)
73$S_4\times C_2$ \( 1 - 1910 T + 5875530055 T^{2} - 8082237219348 T^{3} + 5875530055 p^{5} T^{4} - 1910 p^{10} T^{5} + p^{15} T^{6} \)
79$S_4\times C_2$ \( 1 + 36844 T + 4302241965 T^{2} + 155969878390184 T^{3} + 4302241965 p^{5} T^{4} + 36844 p^{10} T^{5} + p^{15} T^{6} \)
83$S_4\times C_2$ \( 1 + 48796 T + 9666032953 T^{2} + 314032777245928 T^{3} + 9666032953 p^{5} T^{4} + 48796 p^{10} T^{5} + p^{15} T^{6} \)
89$S_4\times C_2$ \( 1 + 188978 T + 25306287767 T^{2} + 2094440123430812 T^{3} + 25306287767 p^{5} T^{4} + 188978 p^{10} T^{5} + p^{15} T^{6} \)
97$S_4\times C_2$ \( 1 - 247526 T + 41254986031 T^{2} - 4400049090000852 T^{3} + 41254986031 p^{5} T^{4} - 247526 p^{10} T^{5} + p^{15} T^{6} \)
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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

−11.13341914292434618718985364345, −10.78224351647095496018801584601, −10.20538566570068564590945717519, −10.18622369971246515459419562955, −9.607877424638736412911168724111, −9.490724777722922210731898760876, −9.374430260570121437577126037473, −8.612606613157344566155336451451, −8.430857834540710248012987246452, −8.347440477958327608953965417041, −7.22066646521201092076179309306, −7.14726641791697984368910402828, −6.52482504148741178646838648247, −6.19802634034931418836389869135, −6.02222134005944540956859917722, −5.60948020799022816143088308016, −5.03436452128696154400275564748, −4.90553161478220412891672427015, −4.70271120020569756700175902464, −3.83698103902245781368943026269, −3.65395604422640982017472634607, −3.36720458660070725424361496679, −2.01705694368360652222090528519, −1.59484573552482972426383062529, −1.29324763946524851849160716602, 0, 0, 0, 1.29324763946524851849160716602, 1.59484573552482972426383062529, 2.01705694368360652222090528519, 3.36720458660070725424361496679, 3.65395604422640982017472634607, 3.83698103902245781368943026269, 4.70271120020569756700175902464, 4.90553161478220412891672427015, 5.03436452128696154400275564748, 5.60948020799022816143088308016, 6.02222134005944540956859917722, 6.19802634034931418836389869135, 6.52482504148741178646838648247, 7.14726641791697984368910402828, 7.22066646521201092076179309306, 8.347440477958327608953965417041, 8.430857834540710248012987246452, 8.612606613157344566155336451451, 9.374430260570121437577126037473, 9.490724777722922210731898760876, 9.607877424638736412911168724111, 10.18622369971246515459419562955, 10.20538566570068564590945717519, 10.78224351647095496018801584601, 11.13341914292434618718985364345

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