Properties

Label 6-8400e3-1.1-c1e3-0-3
Degree $6$
Conductor $592704000000$
Sign $1$
Analytic cond. $301765.$
Root an. cond. $8.18989$
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 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(301765.\)
Root analytic conductor: \(8.18989\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8400} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{12} \cdot 3^{3} \cdot 5^{6} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(30.22679820\)
\(L(\frac12)\) \(\approx\) \(30.22679820\)
\(L(\frac{3}{2})\) 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

−6.90073100734006131579281479251, −6.66788761025284282990784520556, −6.58238788480490738632302788537, −6.24434360217085900286915605009, −5.76122787194553455772750361095, −5.74401165260212325499734563478, −5.72384291173585947344252326255, −5.10679529655927238229789347858, −4.97764309443618997495138710730, −4.88130399781308794652524966600, −4.15323636676385428652050726610, −4.13216278606118092736492517075, −3.93768819095631158822209549290, −3.71960989428145252471578072829, −3.60662316340075041267615170085, −3.38663535161985769080740079420, −2.78410955200357908504867536212, −2.68911387066875200458298158460, −2.55068664036218339342980537262, −1.87298135954012854135112759891, −1.76776781155872865809920997931, −1.63246352950289539847549669438, −1.00861738215017453244413305915, −0.843999041201389317655126401699, −0.830801443534088961792479491938, 0.830801443534088961792479491938, 0.843999041201389317655126401699, 1.00861738215017453244413305915, 1.63246352950289539847549669438, 1.76776781155872865809920997931, 1.87298135954012854135112759891, 2.55068664036218339342980537262, 2.68911387066875200458298158460, 2.78410955200357908504867536212, 3.38663535161985769080740079420, 3.60662316340075041267615170085, 3.71960989428145252471578072829, 3.93768819095631158822209549290, 4.13216278606118092736492517075, 4.15323636676385428652050726610, 4.88130399781308794652524966600, 4.97764309443618997495138710730, 5.10679529655927238229789347858, 5.72384291173585947344252326255, 5.74401165260212325499734563478, 5.76122787194553455772750361095, 6.24434360217085900286915605009, 6.58238788480490738632302788537, 6.66788761025284282990784520556, 6.90073100734006131579281479251

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