Properties

Label 6-7728e3-1.1-c1e3-0-8
Degree $6$
Conductor $461531492352$
Sign $-1$
Analytic cond. $234980.$
Root an. cond. $7.85546$
Motivic weight $1$
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  + 3·3-s − 5-s − 3·7-s + 6·9-s − 11-s + 3·13-s − 3·15-s + 2·17-s − 3·19-s − 9·21-s − 3·23-s − 7·25-s + 10·27-s − 10·29-s − 4·31-s − 3·33-s + 3·35-s − 6·37-s + 9·39-s − 41-s − 13·43-s − 6·45-s − 11·47-s + 6·49-s + 6·51-s − 12·53-s + 55-s + ⋯
L(s)  = 1  + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s − 0.301·11-s + 0.832·13-s − 0.774·15-s + 0.485·17-s − 0.688·19-s − 1.96·21-s − 0.625·23-s − 7/5·25-s + 1.92·27-s − 1.85·29-s − 0.718·31-s − 0.522·33-s + 0.507·35-s − 0.986·37-s + 1.44·39-s − 0.156·41-s − 1.98·43-s − 0.894·45-s − 1.60·47-s + 6/7·49-s + 0.840·51-s − 1.64·53-s + 0.134·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\)
Sign: $-1$
Analytic conductor: \(234980.\)
Root analytic conductor: \(7.85546\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 - T )^{3} \)
7$C_1$ \( ( 1 + T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good5$S_4\times C_2$ \( 1 + T + 8 T^{2} + 6 T^{3} + 8 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + T + 4 T^{2} + 25 T^{3} + 4 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 3 T + 24 T^{2} - 80 T^{3} + 24 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 32 T^{2} - 36 T^{3} + 32 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$C_2$ \( ( 1 + T + p T^{2} )^{3} \)
29$S_4\times C_2$ \( 1 + 10 T + 90 T^{2} + 562 T^{3} + 90 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 4 T + 78 T^{2} + 242 T^{3} + 78 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 6 T + 54 T^{2} + 530 T^{3} + 54 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + T + 2 T^{2} - 327 T^{3} + 2 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 13 T + 148 T^{2} + 1094 T^{3} + 148 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 11 T + 151 T^{2} + 926 T^{3} + 151 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 12 T + 189 T^{2} + 1245 T^{3} + 189 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 12 T + 207 T^{2} + 1389 T^{3} + 207 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 22 T + 337 T^{2} - 3023 T^{3} + 337 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - T + 194 T^{2} - 130 T^{3} + 194 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 7 T + 68 T^{2} + 90 T^{3} + 68 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T + 194 T^{2} - 592 T^{3} + 194 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 8 T + 82 T^{2} - 14 T^{3} + 82 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 10 T + 252 T^{2} + 1642 T^{3} + 252 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 15 T + 324 T^{2} - 2724 T^{3} + 324 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 10 T + 170 T^{2} - 2182 T^{3} + 170 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.58949645603500855536157795902, −7.00921819758640952608140608344, −6.79403890883695540593021040575, −6.76993923470952246257103141829, −6.29582011884631968065643436048, −6.27215051840628555631331855670, −6.14166999140003955803396496304, −5.49474836182993703351749928408, −5.38486727027949971778429960161, −5.19207350813321755905914329741, −5.09014532184322965780287452511, −4.46632165789793993488487958456, −4.28142298240783023979804236981, −3.96059584875605183018811950855, −3.81006397087721952126679609148, −3.65471431556590167478210916768, −3.30633816949483669220615068527, −3.27580504993202616754446560811, −3.04618585280152059349048963970, −2.34783916829201060517154427686, −2.26883984666487826042679093015, −2.22801481779099395791771375818, −1.54729441987632912319477329718, −1.40810761122548944621775598874, −1.23197451438895284923397506425, 0, 0, 0, 1.23197451438895284923397506425, 1.40810761122548944621775598874, 1.54729441987632912319477329718, 2.22801481779099395791771375818, 2.26883984666487826042679093015, 2.34783916829201060517154427686, 3.04618585280152059349048963970, 3.27580504993202616754446560811, 3.30633816949483669220615068527, 3.65471431556590167478210916768, 3.81006397087721952126679609148, 3.96059584875605183018811950855, 4.28142298240783023979804236981, 4.46632165789793993488487958456, 5.09014532184322965780287452511, 5.19207350813321755905914329741, 5.38486727027949971778429960161, 5.49474836182993703351749928408, 6.14166999140003955803396496304, 6.27215051840628555631331855670, 6.29582011884631968065643436048, 6.76993923470952246257103141829, 6.79403890883695540593021040575, 7.00921819758640952608140608344, 7.58949645603500855536157795902

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