Properties

Label 6-4200e3-1.1-c1e3-0-3
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 − 10·13-s − 8·17-s − 2·19-s + 9·21-s − 8·23-s + 10·27-s − 6·29-s − 14·31-s − 6·33-s − 12·37-s − 30·39-s − 10·41-s − 4·43-s + 6·49-s − 24·51-s − 14·53-s − 6·57-s − 8·59-s + 2·61-s + 18·63-s + 16·67-s − 24·69-s − 18·71-s + ⋯
L(s)  = 1  + 1.73·3-s + 1.13·7-s + 2·9-s − 0.603·11-s − 2.77·13-s − 1.94·17-s − 0.458·19-s + 1.96·21-s − 1.66·23-s + 1.92·27-s − 1.11·29-s − 2.51·31-s − 1.04·33-s − 1.97·37-s − 4.80·39-s − 1.56·41-s − 0.609·43-s + 6/7·49-s − 3.36·51-s − 1.92·53-s − 0.794·57-s − 1.04·59-s + 0.256·61-s + 2.26·63-s + 1.95·67-s − 2.88·69-s − 2.13·71-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$S_4\times C_2$ \( 1 + 2 T - 3 T^{2} - 60 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 10 T + 59 T^{2} + 252 T^{3} + 59 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 8 T + 19 T^{2} + 19 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T - 3 T^{2} - 124 T^{3} - 3 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 8 T + 29 T^{2} + 64 T^{3} + 29 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
31$S_4\times C_2$ \( 1 + 14 T + 145 T^{2} + 908 T^{3} + 145 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 12 T + 95 T^{2} + 568 T^{3} + 95 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 143 T^{2} + 812 T^{3} + 143 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 4 T + 81 T^{2} + 280 T^{3} + 81 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
47$C_2$ \( ( 1 + p T^{2} )^{3} \)
53$S_4\times C_2$ \( 1 + 14 T + 211 T^{2} + 1524 T^{3} + 211 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
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$D_{6}$ \( 1 - 2 T - 29 T^{2} - 140 T^{3} - 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 16 T + 3 p T^{2} - 1888 T^{3} + 3 p^{2} T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 18 T + 161 T^{2} + 1204 T^{3} + 161 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 22 T + 319 T^{2} + 3316 T^{3} + 319 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 4 T + 189 T^{2} - 568 T^{3} + 189 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
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$S_4\times C_2$ \( 1 + 6 T + 143 T^{2} + 1300 T^{3} + 143 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 2 T + 231 T^{2} + 188 T^{3} + 231 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.87172370645974286616975032757, −7.48941488940498591763752311124, −7.46795716431259077545486177873, −7.16719823571096004872470322825, −6.85212297635742462706175808599, −6.81088062801532027047112678647, −6.67453230930805001457801325873, −5.94471328865393940324815574248, −5.68549813970385453735718566569, −5.61871155673121720869025127149, −5.14817378601398846576367799781, −4.99464767735618546645588708149, −4.72577105435296754415765324824, −4.39193067773115724582865419534, −4.32601305728432754976549946292, −4.05614402174423736978457445945, −3.47224961349228358669088586777, −3.46473071298810247372840233948, −3.07631829127660821411662706522, −2.64323732261231479945707090360, −2.34755630139913329310073037700, −2.21890120466738899949467778631, −1.76586303225375236870735745231, −1.67319316384251515462886010511, −1.57310305169716197946147723672, 0, 0, 0, 1.57310305169716197946147723672, 1.67319316384251515462886010511, 1.76586303225375236870735745231, 2.21890120466738899949467778631, 2.34755630139913329310073037700, 2.64323732261231479945707090360, 3.07631829127660821411662706522, 3.46473071298810247372840233948, 3.47224961349228358669088586777, 4.05614402174423736978457445945, 4.32601305728432754976549946292, 4.39193067773115724582865419534, 4.72577105435296754415765324824, 4.99464767735618546645588708149, 5.14817378601398846576367799781, 5.61871155673121720869025127149, 5.68549813970385453735718566569, 5.94471328865393940324815574248, 6.67453230930805001457801325873, 6.81088062801532027047112678647, 6.85212297635742462706175808599, 7.16719823571096004872470322825, 7.46795716431259077545486177873, 7.48941488940498591763752311124, 7.87172370645974286616975032757

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