Properties

Label 6-1900e3-1.1-c1e3-0-1
Degree $6$
Conductor $6859000000$
Sign $1$
Analytic cond. $3492.14$
Root an. cond. $3.89507$
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 + 2·7-s − 11-s + 3·13-s + 6·17-s + 3·19-s + 4·21-s + 16·23-s − 5·27-s + 3·29-s − 31-s − 2·33-s − 13·37-s + 6·39-s − 3·41-s + 13·43-s + 9·47-s + 2·49-s + 12·51-s − 53-s + 6·57-s − 6·59-s − 5·61-s + 19·67-s + 32·69-s − 3·71-s + 15·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.755·7-s − 0.301·11-s + 0.832·13-s + 1.45·17-s + 0.688·19-s + 0.872·21-s + 3.33·23-s − 0.962·27-s + 0.557·29-s − 0.179·31-s − 0.348·33-s − 2.13·37-s + 0.960·39-s − 0.468·41-s + 1.98·43-s + 1.31·47-s + 2/7·49-s + 1.68·51-s − 0.137·53-s + 0.794·57-s − 0.781·59-s − 0.640·61-s + 2.32·67-s + 3.85·69-s − 0.356·71-s + 1.75·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(3492.14\)
Root analytic conductor: \(3.89507\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(8.339853235\)
\(L(\frac12)\) \(\approx\) \(8.339853235\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19$C_1$ \( ( 1 - T )^{3} \)
good3$D_{6}$ \( 1 - 2 T + 4 T^{2} - p T^{3} + 4 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
7$D_{6}$ \( 1 - 2 T + 2 T^{2} + 19 T^{3} + 2 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + T + 27 T^{2} + 25 T^{3} + 27 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 3 T + 33 T^{2} - 71 T^{3} + 33 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 6 T + 36 T^{2} - 105 T^{3} + 36 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 16 T + 150 T^{2} - 865 T^{3} + 150 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 3 T + 63 T^{2} - 201 T^{3} + 63 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + T + 53 T^{2} - 47 T^{3} + 53 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 13 T + 161 T^{2} + 1021 T^{3} + 161 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 3 T + 87 T^{2} + 219 T^{3} + 87 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 13 T + 160 T^{2} - 1049 T^{3} + 160 p T^{4} - 13 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 9 T + 45 T^{2} - 225 T^{3} + 45 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + T + 90 T^{2} + 109 T^{3} + 90 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 153 T^{2} + 636 T^{3} + 153 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 5 T + 85 T^{2} + 121 T^{3} + 85 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 19 T + 175 T^{2} - 1223 T^{3} + 175 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 3 T + 180 T^{2} + 399 T^{3} + 180 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 15 T + 261 T^{2} - 2141 T^{3} + 261 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 2 T + 80 T^{2} - 845 T^{3} + 80 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 29 T + 525 T^{2} - 5675 T^{3} + 525 p T^{4} - 29 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 14 T + 258 T^{2} - 2003 T^{3} + 258 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + T + 137 T^{2} + 877 T^{3} + 137 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
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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

−8.238259162773315499045090391235, −7.75731724180906823064219099544, −7.71204780021486835538272686341, −7.62618160401911137852447735612, −7.16599030934889343412760490463, −6.87668065267013255043197134543, −6.85115601380819558202290901603, −6.12361810067307162781741386616, −6.12113002595069626494772234844, −5.71781880653084651999966974181, −5.31266877318578960520081929516, −5.15077996383917758875391134029, −4.87913121563810498962963338356, −4.79452152133517420988190701965, −4.19517924977718085181602622425, −3.76540006699387138889757868476, −3.36068857213726195572919326023, −3.34981108553319591268307388853, −3.21635984610875638109238570633, −2.58735839504172990475993803485, −2.29903910969364766398490487144, −2.02611803913833048515083768739, −1.36228167111732075462598258815, −0.951754256663920820649953696049, −0.78262388696915603577421783986, 0.78262388696915603577421783986, 0.951754256663920820649953696049, 1.36228167111732075462598258815, 2.02611803913833048515083768739, 2.29903910969364766398490487144, 2.58735839504172990475993803485, 3.21635984610875638109238570633, 3.34981108553319591268307388853, 3.36068857213726195572919326023, 3.76540006699387138889757868476, 4.19517924977718085181602622425, 4.79452152133517420988190701965, 4.87913121563810498962963338356, 5.15077996383917758875391134029, 5.31266877318578960520081929516, 5.71781880653084651999966974181, 6.12113002595069626494772234844, 6.12361810067307162781741386616, 6.85115601380819558202290901603, 6.87668065267013255043197134543, 7.16599030934889343412760490463, 7.62618160401911137852447735612, 7.71204780021486835538272686341, 7.75731724180906823064219099544, 8.238259162773315499045090391235

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