Properties

Label 6-567e3-1.1-c1e3-0-0
Degree $6$
Conductor $182284263$
Sign $1$
Analytic cond. $92.8069$
Root an. cond. $2.12779$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·5-s + 3·7-s + 3·8-s − 6·11-s + 3·13-s − 3·17-s + 6·23-s − 3·29-s + 6·31-s − 9·35-s + 15·37-s − 9·40-s + 12·41-s + 12·43-s + 6·49-s − 12·53-s + 18·55-s + 9·56-s + 18·59-s − 3·61-s + 64-s − 9·65-s + 6·67-s + 9·73-s − 18·77-s + 6·79-s − 12·83-s + ⋯
L(s)  = 1  − 1.34·5-s + 1.13·7-s + 1.06·8-s − 1.80·11-s + 0.832·13-s − 0.727·17-s + 1.25·23-s − 0.557·29-s + 1.07·31-s − 1.52·35-s + 2.46·37-s − 1.42·40-s + 1.87·41-s + 1.82·43-s + 6/7·49-s − 1.64·53-s + 2.42·55-s + 1.20·56-s + 2.34·59-s − 0.384·61-s + 1/8·64-s − 1.11·65-s + 0.733·67-s + 1.05·73-s − 2.05·77-s + 0.675·79-s − 1.31·83-s + ⋯

Functional equation

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

Invariants

Degree: \(6\)
Conductor: \(3^{12} \cdot 7^{3}\)
Sign: $1$
Analytic conductor: \(92.8069\)
Root analytic conductor: \(2.12779\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 3^{12} \cdot 7^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(2.050028360\)
\(L(\frac12)\) \(\approx\) \(2.050028360\)
\(L(\frac{3}{2})\) 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)$
bad3 \( 1 \)
7$C_1$ \( ( 1 - T )^{3} \)
good2$D_{6}$ \( 1 - 3 T^{3} + p^{3} T^{6} \)
5$S_4\times C_2$ \( 1 + 3 T + 9 T^{2} + 18 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 6 T + 36 T^{2} + 126 T^{3} + 36 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 3 T + 3 T^{2} + 34 T^{3} + 3 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 3 T + 27 T^{2} + 114 T^{3} + 27 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 9 T^{2} - 56 T^{3} + 9 p T^{4} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 - 6 T + 45 T^{2} - 180 T^{3} + 45 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 3 T + 51 T^{2} + 210 T^{3} + 51 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{3} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{3} \)
41$S_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 912 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 12 T + 84 T^{2} - 380 T^{3} + 84 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 117 T^{2} + 24 T^{3} + 117 p T^{4} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 12 T + 180 T^{2} + 1266 T^{3} + 180 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 18 T + 189 T^{2} - 1572 T^{3} + 189 p T^{4} - 18 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 3 T + 105 T^{2} + 178 T^{3} + 105 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 6 T + 132 T^{2} - 542 T^{3} + 132 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 132 T^{2} + 108 T^{3} + 132 p T^{4} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 9 T + 207 T^{2} - 1298 T^{3} + 207 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 - 6 T + 168 T^{2} - 686 T^{3} + 168 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 12 T + 3 p T^{2} + 1920 T^{3} + 3 p^{2} T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 15 T + 315 T^{2} + 2622 T^{3} + 315 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 12 T + 3 p T^{2} - 2144 T^{3} + 3 p^{2} T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.645581392557475341830787099772, −9.117042056171947074786138601113, −8.991995863000387477360003426770, −8.675807900136106353411101632154, −8.049930774046966205195793375803, −7.997790217070732317057995708772, −7.88560486120336934818194137283, −7.55962259273529413884355354640, −7.51649042831801620093837149522, −6.89238860850333395018729279396, −6.62036596123719874308845812674, −6.17642581414690064612506196135, −5.75756433197239491036309687589, −5.45599049509064150458631604132, −5.03389189316331404711274933554, −4.82726169756478738766241634704, −4.24350905981333780661521834035, −4.17067308450487679386062782940, −4.03627440016763742496060989068, −3.25618973559653280192275619474, −2.70814742136531414936084382173, −2.51032438587698456512283239726, −1.97327481307346946432861525220, −1.13520287727974584344038223712, −0.69247686109260914487240933215, 0.69247686109260914487240933215, 1.13520287727974584344038223712, 1.97327481307346946432861525220, 2.51032438587698456512283239726, 2.70814742136531414936084382173, 3.25618973559653280192275619474, 4.03627440016763742496060989068, 4.17067308450487679386062782940, 4.24350905981333780661521834035, 4.82726169756478738766241634704, 5.03389189316331404711274933554, 5.45599049509064150458631604132, 5.75756433197239491036309687589, 6.17642581414690064612506196135, 6.62036596123719874308845812674, 6.89238860850333395018729279396, 7.51649042831801620093837149522, 7.55962259273529413884355354640, 7.88560486120336934818194137283, 7.997790217070732317057995708772, 8.049930774046966205195793375803, 8.675807900136106353411101632154, 8.991995863000387477360003426770, 9.117042056171947074786138601113, 9.645581392557475341830787099772

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