Properties

Label 6-4200e3-1.1-c1e3-0-2
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 $3$

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: induced by $\chi_{4200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
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

−7.69293959077972303293268548239, −7.42291578674046365785240599840, −7.38850968380835397340339067276, −6.95034170658560005106283338444, −6.69258791855586166828312233604, −6.44486560519779593101161562527, −6.41821501354579640339154606050, −5.97838545750475262426520244071, −5.87707986030733065774471841820, −5.84712628176337039058763954526, −5.10547079790514050387330302316, −5.09378308010713759211101899925, −5.08595788298688738686373603200, −4.41637012704100646955585425713, −4.24454268024847968585149365933, −4.19665787990758347454261913576, −3.59216739070791161437655696357, −3.51962558557083616282144112705, −3.29012444072229198334689558741, −2.56885792051629769245362040138, −2.40917284692859518309199568995, −2.36015385608100650414984528349, −1.33179347039825393918165419925, −1.30640547882914576045480553794, −1.24538636827853472747619504811, 0, 0, 0, 1.24538636827853472747619504811, 1.30640547882914576045480553794, 1.33179347039825393918165419925, 2.36015385608100650414984528349, 2.40917284692859518309199568995, 2.56885792051629769245362040138, 3.29012444072229198334689558741, 3.51962558557083616282144112705, 3.59216739070791161437655696357, 4.19665787990758347454261913576, 4.24454268024847968585149365933, 4.41637012704100646955585425713, 5.08595788298688738686373603200, 5.09378308010713759211101899925, 5.10547079790514050387330302316, 5.84712628176337039058763954526, 5.87707986030733065774471841820, 5.97838545750475262426520244071, 6.41821501354579640339154606050, 6.44486560519779593101161562527, 6.69258791855586166828312233604, 6.95034170658560005106283338444, 7.38850968380835397340339067276, 7.42291578674046365785240599840, 7.69293959077972303293268548239

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