Properties

Label 6-7800e3-1.1-c1e3-0-13
Degree $6$
Conductor $474552000000$
Sign $-1$
Analytic cond. $241610.$
Root an. cond. $7.89197$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·3-s + 6·9-s − 6·11-s + 3·13-s − 4·17-s − 2·19-s − 10·23-s + 10·27-s − 10·29-s + 12·31-s − 18·33-s + 6·37-s + 9·39-s − 10·41-s + 4·43-s − 16·47-s − 21·49-s − 12·51-s − 10·53-s − 6·57-s − 6·59-s + 2·61-s − 4·67-s − 30·69-s + 8·71-s − 6·73-s + 20·79-s + ⋯
L(s)  = 1  + 1.73·3-s + 2·9-s − 1.80·11-s + 0.832·13-s − 0.970·17-s − 0.458·19-s − 2.08·23-s + 1.92·27-s − 1.85·29-s + 2.15·31-s − 3.13·33-s + 0.986·37-s + 1.44·39-s − 1.56·41-s + 0.609·43-s − 2.33·47-s − 3·49-s − 1.68·51-s − 1.37·53-s − 0.794·57-s − 0.781·59-s + 0.256·61-s − 0.488·67-s − 3.61·69-s + 0.949·71-s − 0.702·73-s + 2.25·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(241610.\)
Root analytic conductor: \(7.89197\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{7800} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 3^{3} \cdot 5^{6} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
13$C_1$ \( ( 1 - T )^{3} \)
good7$C_2$ \( ( 1 + p T^{2} )^{3} \)
11$S_4\times C_2$ \( 1 + 6 T + 35 T^{2} + 112 T^{3} + 35 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 31 T^{2} + 72 T^{3} + 31 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 2 T + 33 T^{2} + 92 T^{3} + 33 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 10 T + 77 T^{2} + 380 T^{3} + 77 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 10 T + 43 T^{2} + 108 T^{3} + 43 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 12 T + 101 T^{2} - 584 T^{3} + 101 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 6 T + 83 T^{2} - 308 T^{3} + 83 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 10 T + 89 T^{2} + 488 T^{3} + 89 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 4 T + 109 T^{2} - 280 T^{3} + 109 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 16 T + 175 T^{2} + 1248 T^{3} + 175 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 10 T + 115 T^{2} + 588 T^{3} + 115 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 99 T^{2} + 752 T^{3} + 99 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$D_{6}$ \( 1 - 2 T + 151 T^{2} - 212 T^{3} + 151 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 4 T + 73 T^{2} + 280 T^{3} + 73 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 8 T + 215 T^{2} - 1072 T^{3} + 215 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 6 T + 71 T^{2} + 52 T^{3} + 71 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 20 T + 345 T^{2} - 3320 T^{3} + 345 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 4 T + 187 T^{2} + 432 T^{3} + 187 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 22 T + 361 T^{2} + 3672 T^{3} + 361 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
97$C_2$ \( ( 1 + 6 T + p T^{2} )^{3} \)
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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

−7.67215588937917696884675765736, −6.87625954787913292142825694593, −6.77953939702828842126594776782, −6.77803605636580761143779899533, −6.27699128499688535294274464680, −6.25977116967052907754952054886, −6.14137519558951694881539820548, −5.40910274308241159375831653547, −5.36761156507277464225464033536, −5.33471249725517386297250747019, −4.67905124359888426348459750668, −4.60479022217698224353097100293, −4.56001953857520854050563700875, −3.94130844640860387594086373671, −3.84425527641448233382690925638, −3.82001045538858944954337429609, −3.22502332833715182511292437417, −3.01254834826697086666788039597, −2.95804649553138258896601839949, −2.39642490723420324740181388671, −2.38455195555643670385679375536, −2.14738454343177271451854765155, −1.49965898496689832250859761779, −1.38887103708305694508247303011, −1.38319982783270724792277001444, 0, 0, 0, 1.38319982783270724792277001444, 1.38887103708305694508247303011, 1.49965898496689832250859761779, 2.14738454343177271451854765155, 2.38455195555643670385679375536, 2.39642490723420324740181388671, 2.95804649553138258896601839949, 3.01254834826697086666788039597, 3.22502332833715182511292437417, 3.82001045538858944954337429609, 3.84425527641448233382690925638, 3.94130844640860387594086373671, 4.56001953857520854050563700875, 4.60479022217698224353097100293, 4.67905124359888426348459750668, 5.33471249725517386297250747019, 5.36761156507277464225464033536, 5.40910274308241159375831653547, 6.14137519558951694881539820548, 6.25977116967052907754952054886, 6.27699128499688535294274464680, 6.77803605636580761143779899533, 6.77953939702828842126594776782, 6.87625954787913292142825694593, 7.67215588937917696884675765736

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