Properties

Label 6-4009e3-1.1-c1e3-0-0
Degree $6$
Conductor $64432972729$
Sign $-1$
Analytic cond. $32804.9$
Root an. cond. $5.65791$
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  + 2·2-s − 2·3-s − 4-s − 3·5-s − 4·6-s + 3·7-s − 5·8-s − 4·9-s − 6·10-s + 11-s + 2·12-s + 6·14-s + 6·15-s − 16-s − 2·17-s − 8·18-s + 3·19-s + 3·20-s − 6·21-s + 2·22-s + 2·23-s + 10·24-s − 2·25-s + 13·27-s − 3·28-s + 6·29-s + 12·30-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s − 1/2·4-s − 1.34·5-s − 1.63·6-s + 1.13·7-s − 1.76·8-s − 4/3·9-s − 1.89·10-s + 0.301·11-s + 0.577·12-s + 1.60·14-s + 1.54·15-s − 1/4·16-s − 0.485·17-s − 1.88·18-s + 0.688·19-s + 0.670·20-s − 1.30·21-s + 0.426·22-s + 0.417·23-s + 2.04·24-s − 2/5·25-s + 2.50·27-s − 0.566·28-s + 1.11·29-s + 2.19·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{3} \cdot 211^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(19^{3} \cdot 211^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(19^{3} \cdot 211^{3}\)
Sign: $-1$
Analytic conductor: \(32804.9\)
Root analytic conductor: \(5.65791\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 19^{3} \cdot 211^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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)$
bad19$C_1$ \( ( 1 - T )^{3} \)
211$C_1$ \( ( 1 - T )^{3} \)
good2$A_4\times C_2$ \( 1 - p T + 5 T^{2} - 7 T^{3} + 5 p T^{4} - p^{3} T^{5} + p^{3} T^{6} \)
3$A_4\times C_2$ \( 1 + 2 T + 8 T^{2} + 11 T^{3} + 8 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
5$A_4\times C_2$ \( 1 + 3 T + 11 T^{2} + 17 T^{3} + 11 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
7$A_4\times C_2$ \( 1 - 3 T + 17 T^{2} - 29 T^{3} + 17 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
11$A_4\times C_2$ \( 1 - T + 24 T^{2} - 21 T^{3} + 24 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
13$C_2$ \( ( 1 + p T^{2} )^{3} \)
17$A_4\times C_2$ \( 1 + 2 T + 50 T^{2} + 67 T^{3} + 50 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
23$A_4\times C_2$ \( 1 - 2 T + 68 T^{2} - 91 T^{3} + 68 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
29$A_4\times C_2$ \( 1 - 6 T + 92 T^{2} - 335 T^{3} + 92 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
31$A_4\times C_2$ \( 1 - 5 T + 57 T^{2} - 143 T^{3} + 57 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$A_4\times C_2$ \( 1 - 9 T + 47 T^{2} - 217 T^{3} + 47 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
41$A_4\times C_2$ \( 1 + 7 T + 95 T^{2} + 581 T^{3} + 95 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
43$A_4\times C_2$ \( 1 + T + 120 T^{2} + 85 T^{3} + 120 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
47$A_4\times C_2$ \( 1 + 3 T + 53 T^{2} + 479 T^{3} + 53 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
53$A_4\times C_2$ \( 1 + 27 T + 395 T^{2} + 3535 T^{3} + 395 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
59$A_4\times C_2$ \( 1 + 2 T + 148 T^{2} + 249 T^{3} + 148 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
61$A_4\times C_2$ \( 1 - 4 T + 179 T^{2} - 480 T^{3} + 179 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
67$A_4\times C_2$ \( 1 - 8 T + 45 T^{2} + 280 T^{3} + 45 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
71$A_4\times C_2$ \( 1 + T + 127 T^{2} - 195 T^{3} + 127 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
73$A_4\times C_2$ \( 1 - 12 T + 246 T^{2} - 1739 T^{3} + 246 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
79$A_4\times C_2$ \( 1 + 33 T + 593 T^{2} + 6475 T^{3} + 593 p T^{4} + 33 p^{2} T^{5} + p^{3} T^{6} \)
83$A_4\times C_2$ \( 1 + 7 T + 221 T^{2} + 959 T^{3} + 221 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
89$A_4\times C_2$ \( 1 - 4 T - 24 T^{2} + 1641 T^{3} - 24 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
97$A_4\times C_2$ \( 1 + 7 T + 3 p T^{2} + 1309 T^{3} + 3 p^{2} T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
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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

−7.959542409236452173542121465243, −7.68811880175836466272782701661, −7.47810080104916496366525281822, −6.85249638898217771498435458475, −6.74464575639568127300230176911, −6.54178302240384191757751189911, −6.20920864070520855965750703770, −6.03677605859355143802374927355, −5.71057130631797946951601385344, −5.48294246294033628407558444557, −5.11827668246389851598242479963, −5.08002618641658441607554371738, −4.83120700717509920557320838683, −4.58562947134007457113370804820, −4.46103528426471747673282487134, −4.10741939296364920882963025359, −3.79572104337068078377712193302, −3.51802446143627229188944747516, −3.40235882250066185805968147829, −2.82597799379438794977861726925, −2.61694061756832372359671336598, −2.56572384437823008000571798873, −1.49501662985074360649748489447, −1.42463846276997048220395625802, −1.04242084415200452655648978525, 0, 0, 0, 1.04242084415200452655648978525, 1.42463846276997048220395625802, 1.49501662985074360649748489447, 2.56572384437823008000571798873, 2.61694061756832372359671336598, 2.82597799379438794977861726925, 3.40235882250066185805968147829, 3.51802446143627229188944747516, 3.79572104337068078377712193302, 4.10741939296364920882963025359, 4.46103528426471747673282487134, 4.58562947134007457113370804820, 4.83120700717509920557320838683, 5.08002618641658441607554371738, 5.11827668246389851598242479963, 5.48294246294033628407558444557, 5.71057130631797946951601385344, 6.03677605859355143802374927355, 6.20920864070520855965750703770, 6.54178302240384191757751189911, 6.74464575639568127300230176911, 6.85249638898217771498435458475, 7.47810080104916496366525281822, 7.68811880175836466272782701661, 7.959542409236452173542121465243

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