Properties

Label 6-4200e3-1.1-c1e3-0-1
Degree $6$
Conductor $74088000000$
Sign $1$
Analytic cond. $37720.6$
Root an. cond. $5.79112$
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  + 3·3-s + 3·7-s + 6·9-s + 2·11-s + 2·13-s + 2·19-s + 9·21-s + 8·23-s + 10·27-s + 2·29-s − 2·31-s + 6·33-s − 4·37-s + 6·39-s + 14·41-s + 12·43-s + 8·47-s + 6·49-s + 14·53-s + 6·57-s + 8·59-s + 2·61-s + 18·63-s + 24·69-s − 6·71-s + 6·73-s + 6·77-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.13·7-s + 2·9-s + 0.603·11-s + 0.554·13-s + 0.458·19-s + 1.96·21-s + 1.66·23-s + 1.92·27-s + 0.371·29-s − 0.359·31-s + 1.04·33-s − 0.657·37-s + 0.960·39-s + 2.18·41-s + 1.82·43-s + 1.16·47-s + 6/7·49-s + 1.92·53-s + 0.794·57-s + 1.04·59-s + 0.256·61-s + 2.26·63-s + 2.88·69-s − 0.712·71-s + 0.702·73-s + 0.683·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(37720.6\)
Root analytic conductor: \(5.79112\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(18.85172736\)
\(L(\frac12)\) \(\approx\) \(18.85172736\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
5 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
good11$S_4\times C_2$ \( 1 - 2 T + 13 T^{2} - 36 T^{3} + 13 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
13$D_{6}$ \( 1 - 2 T + 27 T^{2} - 44 T^{3} + 27 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 35 T^{2} + 16 T^{3} + 35 p T^{4} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 2 T + 45 T^{2} - 68 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 8 T + 77 T^{2} - 352 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
29$D_{6}$ \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 81 T^{2} + 116 T^{3} + 81 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$D_{6}$ \( 1 + 4 T + 63 T^{2} + 232 T^{3} + 63 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 14 T + 175 T^{2} - 1188 T^{3} + 175 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 12 T + 113 T^{2} - 712 T^{3} + 113 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 8 T + 77 T^{2} - 496 T^{3} + 77 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 14 T + 131 T^{2} - 1012 T^{3} + 131 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 8 T + 113 T^{2} - 688 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$C_2$ \( ( 1 + p T^{2} )^{3} \)
71$S_4\times C_2$ \( 1 + 6 T + 113 T^{2} + 1052 T^{3} + 113 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 6 T + 95 T^{2} - 116 T^{3} + 95 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 20 T + 317 T^{2} + 3096 T^{3} + 317 p T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 8 T + 185 T^{2} - 1072 T^{3} + 185 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 14 T + 295 T^{2} + 2372 T^{3} + 295 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.49830807320932891030685331379, −7.20344651368272013663050767288, −7.10681020642436931139586929943, −7.10666109646303555195974130726, −6.27487560566030994595059401690, −6.23355338228539403082611537164, −6.17785235792307531220184315294, −5.53144955574465468595613735524, −5.39863132398323266778724073630, −5.29323806082654797340945239141, −4.66835388354211714474894872135, −4.53803433038942541686713975712, −4.47582151153130186384050806480, −3.87401323785576461394572541035, −3.81754461601042809758104718046, −3.68926053109828767532394821110, −3.16910756899205535055321063300, −2.80635499180977979136856583914, −2.74754745821256026430320461789, −2.20880213122624788908194740924, −2.11990587021749249498655956296, −1.75049036367339217763819388622, −1.13363329568442560291432284888, −0.890511744380821270566633353663, −0.840922612463377632785879951326, 0.840922612463377632785879951326, 0.890511744380821270566633353663, 1.13363329568442560291432284888, 1.75049036367339217763819388622, 2.11990587021749249498655956296, 2.20880213122624788908194740924, 2.74754745821256026430320461789, 2.80635499180977979136856583914, 3.16910756899205535055321063300, 3.68926053109828767532394821110, 3.81754461601042809758104718046, 3.87401323785576461394572541035, 4.47582151153130186384050806480, 4.53803433038942541686713975712, 4.66835388354211714474894872135, 5.29323806082654797340945239141, 5.39863132398323266778724073630, 5.53144955574465468595613735524, 6.17785235792307531220184315294, 6.23355338228539403082611537164, 6.27487560566030994595059401690, 7.10666109646303555195974130726, 7.10681020642436931139586929943, 7.20344651368272013663050767288, 7.49830807320932891030685331379

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