Properties

Label 6-2320e3-1.1-c1e3-0-0
Degree $6$
Conductor $12487168000$
Sign $1$
Analytic cond. $6357.63$
Root an. cond. $4.30410$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 3·5-s + 4·7-s + 3·9-s − 8·11-s − 2·13-s + 6·15-s + 2·17-s − 8·19-s − 8·21-s + 6·25-s − 4·27-s − 3·29-s − 4·31-s + 16·33-s − 12·35-s − 10·37-s + 4·39-s + 10·41-s − 14·43-s − 9·45-s + 10·47-s − 49-s − 4·51-s + 10·53-s + 24·55-s + 16·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 1.34·5-s + 1.51·7-s + 9-s − 2.41·11-s − 0.554·13-s + 1.54·15-s + 0.485·17-s − 1.83·19-s − 1.74·21-s + 6/5·25-s − 0.769·27-s − 0.557·29-s − 0.718·31-s + 2.78·33-s − 2.02·35-s − 1.64·37-s + 0.640·39-s + 1.56·41-s − 2.13·43-s − 1.34·45-s + 1.45·47-s − 1/7·49-s − 0.560·51-s + 1.37·53-s + 3.23·55-s + 2.11·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{3} \cdot 29^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 5^{3} \cdot 29^{3}\)
Sign: $1$
Analytic conductor: \(6357.63\)
Root analytic conductor: \(4.30410\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 5^{3} \cdot 29^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(0.3403545337\)
\(L(\frac12)\) \(\approx\) \(0.3403545337\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
29$C_1$ \( ( 1 + T )^{3} \)
good3$D_{6}$ \( 1 + 2 T + T^{2} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 4 T + 17 T^{2} - 52 T^{3} + 17 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 8 T + 45 T^{2} + 164 T^{3} + 45 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T + 19 T^{2} + 28 T^{3} + 19 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 15 T^{2} + 40 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 8 T + 41 T^{2} + 140 T^{3} + 41 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 45 T^{2} + 36 T^{3} + 45 p T^{4} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 284 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 10 T + 59 T^{2} + 232 T^{3} + 59 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 10 T + 135 T^{2} - 796 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
43$D_{6}$ \( 1 + 14 T + 113 T^{2} + 656 T^{3} + 113 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 10 T + 165 T^{2} - 952 T^{3} + 165 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 147 T^{2} - 988 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 8 T + 9 T^{2} - 544 T^{3} + 9 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
61$D_{6}$ \( 1 + 22 T + 299 T^{2} + 2788 T^{3} + 299 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 12 T + 225 T^{2} - 1540 T^{3} + 225 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 189 T^{2} - 992 T^{3} + 189 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 2 T + 139 T^{2} - 32 T^{3} + 139 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 129 T^{2} - 244 T^{3} + 129 p T^{4} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 213 T^{2} + 1884 T^{3} + 213 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 18 T + 279 T^{2} - 3132 T^{3} + 279 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 38 T + 763 T^{2} + 9280 T^{3} + 763 p T^{4} + 38 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.924794176140012792945998319896, −7.65796157152496869288335528901, −7.57298361103815706600003024442, −7.15321636092874024033188435453, −7.14482162110104652852548011775, −6.78941942027129863481242153177, −6.45634996574501882516918527172, −5.86590325970201972446550320461, −5.78514484794685701332615633680, −5.69206629916451759272315232861, −5.17465749029928788139774024315, −5.04809027502783838506796251903, −4.65491590974454258454652097099, −4.64305386842206180476554494116, −4.20613938935334327910666933307, −4.19656326550927076822799169220, −3.31772887986094420487517294754, −3.31733148312071953761442195931, −3.11792587591189350917308135677, −2.22720248327096443703740557520, −2.11303724707031596704706335650, −1.97303469333231774575502817397, −1.31277044420626135305584383448, −0.62768833805266549282175939740, −0.21395557027262104424350486756, 0.21395557027262104424350486756, 0.62768833805266549282175939740, 1.31277044420626135305584383448, 1.97303469333231774575502817397, 2.11303724707031596704706335650, 2.22720248327096443703740557520, 3.11792587591189350917308135677, 3.31733148312071953761442195931, 3.31772887986094420487517294754, 4.19656326550927076822799169220, 4.20613938935334327910666933307, 4.64305386842206180476554494116, 4.65491590974454258454652097099, 5.04809027502783838506796251903, 5.17465749029928788139774024315, 5.69206629916451759272315232861, 5.78514484794685701332615633680, 5.86590325970201972446550320461, 6.45634996574501882516918527172, 6.78941942027129863481242153177, 7.14482162110104652852548011775, 7.15321636092874024033188435453, 7.57298361103815706600003024442, 7.65796157152496869288335528901, 7.924794176140012792945998319896

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