Properties

Label 6-6080e3-1.1-c1e3-0-12
Degree $6$
Conductor $224755712000$
Sign $-1$
Analytic cond. $114430.$
Root an. cond. $6.96771$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·5-s + 7-s − 2·9-s − 5·13-s + 3·15-s + 5·17-s + 3·19-s − 21-s + 23-s + 6·25-s + 3·27-s − 17·29-s − 2·31-s − 3·35-s − 8·37-s + 5·39-s + 6·41-s − 10·43-s + 6·45-s − 4·47-s − 8·49-s − 5·51-s − 5·53-s − 3·57-s + 15·59-s − 8·61-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 0.377·7-s − 2/3·9-s − 1.38·13-s + 0.774·15-s + 1.21·17-s + 0.688·19-s − 0.218·21-s + 0.208·23-s + 6/5·25-s + 0.577·27-s − 3.15·29-s − 0.359·31-s − 0.507·35-s − 1.31·37-s + 0.800·39-s + 0.937·41-s − 1.52·43-s + 0.894·45-s − 0.583·47-s − 8/7·49-s − 0.700·51-s − 0.686·53-s − 0.397·57-s + 1.95·59-s − 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 5^{3} \cdot 19^{3}\)
Sign: $-1$
Analytic conductor: \(114430.\)
Root analytic conductor: \(6.96771\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{18} \cdot 5^{3} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
19$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + T + p T^{2} + 2 T^{3} + p^{2} T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - T + 9 T^{2} + 2 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
11$C_2$ \( ( 1 + p T^{2} )^{3} \)
13$S_4\times C_2$ \( 1 + 5 T + 41 T^{2} + 126 T^{3} + 41 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 5 T + 47 T^{2} - 166 T^{3} + 47 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - T + 37 T^{2} + 18 T^{3} + 37 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 17 T + 171 T^{2} + 1110 T^{3} + 171 p T^{4} + 17 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 45 T^{2} - 4 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 8 T + 3 p T^{2} + 584 T^{3} + 3 p^{2} T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 6 T + 71 T^{2} - 436 T^{3} + 71 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 10 T + 137 T^{2} + 828 T^{3} + 137 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 4 T + 61 T^{2} + 504 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 5 T + 117 T^{2} + 582 T^{3} + 117 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 15 T + 149 T^{2} - 986 T^{3} + 149 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 8 T + 179 T^{2} + 912 T^{3} + 179 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 3 T + 23 T^{2} + 650 T^{3} + 23 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 4 T + 21 T^{2} + 456 T^{3} + 21 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 3 T + 119 T^{2} - 10 p T^{3} + 119 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 2 T + 213 T^{2} + 284 T^{3} + 213 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 10 T + 233 T^{2} + 1628 T^{3} + 233 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 6 T + 215 T^{2} - 884 T^{3} + 215 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 4 T + 187 T^{2} + 1072 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.56496545962264884204581112343, −7.41190789645255998133076125893, −7.07826922301391166545448117442, −6.96099031918429608342857010908, −6.46726450129424995994431165013, −6.26931214440018083995467194920, −6.15242330451753197339477091521, −5.72153221276803081070868756728, −5.39507727686279157605254544302, −5.29088063357851811655010305081, −5.12694218986095256182019630689, −4.94258402501710566529610742096, −4.72540219891903678359645411278, −4.28020131384362837639977187223, −3.90543572145978456113344766349, −3.83219110483622820754599308201, −3.44934310784228108646805152838, −3.26448006152997898078047609718, −3.16302080802291365498930980884, −2.63899922200150773136927817504, −2.31364364039494870537115966367, −2.00681994419328283279841848353, −1.69130919022561609280655889687, −1.13876493245878919327131465618, −1.01712521677649612327528413839, 0, 0, 0, 1.01712521677649612327528413839, 1.13876493245878919327131465618, 1.69130919022561609280655889687, 2.00681994419328283279841848353, 2.31364364039494870537115966367, 2.63899922200150773136927817504, 3.16302080802291365498930980884, 3.26448006152997898078047609718, 3.44934310784228108646805152838, 3.83219110483622820754599308201, 3.90543572145978456113344766349, 4.28020131384362837639977187223, 4.72540219891903678359645411278, 4.94258402501710566529610742096, 5.12694218986095256182019630689, 5.29088063357851811655010305081, 5.39507727686279157605254544302, 5.72153221276803081070868756728, 6.15242330451753197339477091521, 6.26931214440018083995467194920, 6.46726450129424995994431165013, 6.96099031918429608342857010908, 7.07826922301391166545448117442, 7.41190789645255998133076125893, 7.56496545962264884204581112343

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