Properties

Label 6-920e3-1.1-c1e3-0-2
Degree $6$
Conductor $778688000$
Sign $1$
Analytic cond. $396.455$
Root an. cond. $2.71039$
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  + 2·3-s + 3·5-s + 7·7-s − 9-s + 3·11-s + 6·13-s + 6·15-s − 5·17-s + 7·19-s + 14·21-s − 3·23-s + 6·25-s − 7·27-s − 29-s + 10·31-s + 6·33-s + 21·35-s + 2·37-s + 12·39-s − 10·41-s + 12·43-s − 3·45-s + 47-s + 17·49-s − 10·51-s + 10·53-s + 9·55-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.34·5-s + 2.64·7-s − 1/3·9-s + 0.904·11-s + 1.66·13-s + 1.54·15-s − 1.21·17-s + 1.60·19-s + 3.05·21-s − 0.625·23-s + 6/5·25-s − 1.34·27-s − 0.185·29-s + 1.79·31-s + 1.04·33-s + 3.54·35-s + 0.328·37-s + 1.92·39-s − 1.56·41-s + 1.82·43-s − 0.447·45-s + 0.145·47-s + 17/7·49-s − 1.40·51-s + 1.37·53-s + 1.21·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{3} \cdot 23^{3}\)
Sign: $1$
Analytic conductor: \(396.455\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(9.888592342\)
\(L(\frac12)\) \(\approx\) \(9.888592342\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
23$C_1$ \( ( 1 + T )^{3} \)
good3$S_4\times C_2$ \( 1 - 2 T + 5 T^{2} - 5 T^{3} + 5 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - p T + 32 T^{2} - 102 T^{3} + 32 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 3 T + 32 T^{2} - 64 T^{3} + 32 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 6 T + 47 T^{2} - 157 T^{3} + 47 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 5 T + 20 T^{2} + 22 T^{3} + 20 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 7 T + 46 T^{2} - 160 T^{3} + 46 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + T - 3 T^{2} - 90 T^{3} - 3 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 10 T + 107 T^{2} - 567 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
37$D_{6}$ \( 1 - 2 T - 17 T^{2} + 364 T^{3} - 17 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 75 T^{2} + 339 T^{3} + 75 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 12 T + 113 T^{2} - 776 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - T + 135 T^{2} - 90 T^{3} + 135 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 107 T^{2} - 1052 T^{3} + 107 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 10 T + 53 T^{2} + 4 T^{3} + 53 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 13 T + 154 T^{2} + 1372 T^{3} + 154 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 6 T + 41 T^{2} - 380 T^{3} + 41 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 10 T + 113 T^{2} - 545 T^{3} + 113 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 7 T + 185 T^{2} - 786 T^{3} + 185 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 14 T + 253 T^{2} - 1988 T^{3} + 253 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 16 T + 309 T^{2} - 2688 T^{3} + 309 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 20 T + 299 T^{2} + 3304 T^{3} + 299 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 5 T + 268 T^{2} - 914 T^{3} + 268 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
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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

−9.143048666239080908928426854935, −8.538055073775153864000519605325, −8.453733484457412234816126566720, −8.263225076565103194560007740773, −7.88629434486098892833799473422, −7.81408982564360192177338553706, −7.47847645189270186122249680242, −6.74382005392414375877452518795, −6.67094484596823662163525989527, −6.52533072433268382376182278651, −5.95139527898640990195929300550, −5.60806728842974548021111471315, −5.55072469656105430894790900652, −5.03608958648837442145271889558, −4.88392995796946947479376817620, −4.39503789783715611643354618660, −4.08597009696423109799559802424, −3.71130921983602377177751038904, −3.45116035649604447973039219012, −2.69527865241564531053553556130, −2.52232716518289051615095279613, −2.26003742420710416231909558645, −1.57397861093359029755109714430, −1.36312686375066251947537413716, −1.08525252440329187717439648106, 1.08525252440329187717439648106, 1.36312686375066251947537413716, 1.57397861093359029755109714430, 2.26003742420710416231909558645, 2.52232716518289051615095279613, 2.69527865241564531053553556130, 3.45116035649604447973039219012, 3.71130921983602377177751038904, 4.08597009696423109799559802424, 4.39503789783715611643354618660, 4.88392995796946947479376817620, 5.03608958648837442145271889558, 5.55072469656105430894790900652, 5.60806728842974548021111471315, 5.95139527898640990195929300550, 6.52533072433268382376182278651, 6.67094484596823662163525989527, 6.74382005392414375877452518795, 7.47847645189270186122249680242, 7.81408982564360192177338553706, 7.88629434486098892833799473422, 8.263225076565103194560007740773, 8.453733484457412234816126566720, 8.538055073775153864000519605325, 9.143048666239080908928426854935

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