Properties

Label 6-3864e3-1.1-c1e3-0-0
Degree $6$
Conductor $57691436544$
Sign $1$
Analytic cond. $29372.6$
Root an. cond. $5.55465$
Motivic weight $1$
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  − 3·3-s + 5-s + 3·7-s + 6·9-s − 2·11-s − 3·13-s − 3·15-s + 2·17-s − 8·19-s − 9·21-s + 3·23-s − 7·25-s − 10·27-s − 2·29-s + 10·31-s + 6·33-s + 3·35-s + 12·37-s + 9·39-s + 16·41-s − 9·43-s + 6·45-s + 14·47-s + 6·49-s − 6·51-s + 15·53-s − 2·55-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 1.13·7-s + 2·9-s − 0.603·11-s − 0.832·13-s − 0.774·15-s + 0.485·17-s − 1.83·19-s − 1.96·21-s + 0.625·23-s − 7/5·25-s − 1.92·27-s − 0.371·29-s + 1.79·31-s + 1.04·33-s + 0.507·35-s + 1.97·37-s + 1.44·39-s + 2.49·41-s − 1.37·43-s + 0.894·45-s + 2.04·47-s + 6/7·49-s − 0.840·51-s + 2.06·53-s − 0.269·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(29372.6\)
Root analytic conductor: \(5.55465\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 7^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.658261366\)
\(L(\frac12)\) \(\approx\) \(2.658261366\)
\(L(\frac{3}{2})\) 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$S_4\times C_2$ \( 1 - T + 8 T^{2} - 4 T^{3} + 8 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 2 T + 14 T^{2} + 8 T^{3} + 14 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 3 T + 28 T^{2} + 84 T^{3} + 28 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 30 T^{2} - 50 T^{3} + 30 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 8 T + 56 T^{2} + 260 T^{3} + 56 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 66 T^{2} + 98 T^{3} + 66 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 10 T + 106 T^{2} - 612 T^{3} + 106 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 12 T + 76 T^{2} - 334 T^{3} + 76 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 16 T + 188 T^{2} - 1378 T^{3} + 188 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 9 T + 142 T^{2} + 778 T^{3} + 142 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 14 T + 41 T^{2} + 156 T^{3} + 41 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 15 T + 118 T^{2} - 668 T^{3} + 118 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 11 T + 152 T^{2} + 950 T^{3} + 152 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 19 T + 168 T^{2} - 1112 T^{3} + 168 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 13 T + 192 T^{2} + 1698 T^{3} + 192 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 13 T + 224 T^{2} + 1822 T^{3} + 224 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 10 T + 122 T^{2} + 782 T^{3} + 122 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 20 T + 348 T^{2} + 3304 T^{3} + 348 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 2 T + 230 T^{2} - 296 T^{3} + 230 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 17 T + 318 T^{2} - 2860 T^{3} + 318 p T^{4} - 17 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 20 T + 404 T^{2} - 4018 T^{3} + 404 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.51293940503012996497188973714, −7.35585750057349629989738233887, −7.12693310656197473349340542756, −6.68558561185794418910135540835, −6.22588977031524097457967343353, −6.17126849484420323891001682887, −6.12935841072089265514384292606, −5.72113923977248593952019455132, −5.55667312788327202223514801582, −5.36449216465488461637718979764, −4.91352583231101600923195519127, −4.69933507342328134389929974256, −4.58365576941791162205047919070, −4.09525262321783458466288721898, −4.09187102363093753131003630835, −4.02748389938973003270379376047, −3.12176993546501503498199484947, −2.84319816016153420059265551106, −2.65444411053154570879551570113, −2.16905885779994197814490936088, −1.87585291800461819990118247868, −1.77315252709323827931719729863, −1.02826289734631684428068665372, −0.62376525087044294524632964602, −0.55777531738839565009568064146, 0.55777531738839565009568064146, 0.62376525087044294524632964602, 1.02826289734631684428068665372, 1.77315252709323827931719729863, 1.87585291800461819990118247868, 2.16905885779994197814490936088, 2.65444411053154570879551570113, 2.84319816016153420059265551106, 3.12176993546501503498199484947, 4.02748389938973003270379376047, 4.09187102363093753131003630835, 4.09525262321783458466288721898, 4.58365576941791162205047919070, 4.69933507342328134389929974256, 4.91352583231101600923195519127, 5.36449216465488461637718979764, 5.55667312788327202223514801582, 5.72113923977248593952019455132, 6.12935841072089265514384292606, 6.17126849484420323891001682887, 6.22588977031524097457967343353, 6.68558561185794418910135540835, 7.12693310656197473349340542756, 7.35585750057349629989738233887, 7.51293940503012996497188973714

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