Properties

Label 6-8550e3-1.1-c1e3-0-9
Degree $6$
Conductor $625026375000$
Sign $1$
Analytic cond. $318221.$
Root an. cond. $8.26269$
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  + 3·2-s + 6·4-s + 2·7-s + 10·8-s + 5·11-s + 4·13-s + 6·14-s + 15·16-s + 2·17-s − 3·19-s + 15·22-s − 23-s + 12·26-s + 12·28-s + 5·29-s + 5·31-s + 21·32-s + 6·34-s + 4·37-s − 9·38-s + 10·41-s + 2·43-s + 30·44-s − 3·46-s + 4·47-s − 11·49-s + 24·52-s + ⋯
L(s)  = 1  + 2.12·2-s + 3·4-s + 0.755·7-s + 3.53·8-s + 1.50·11-s + 1.10·13-s + 1.60·14-s + 15/4·16-s + 0.485·17-s − 0.688·19-s + 3.19·22-s − 0.208·23-s + 2.35·26-s + 2.26·28-s + 0.928·29-s + 0.898·31-s + 3.71·32-s + 1.02·34-s + 0.657·37-s − 1.45·38-s + 1.56·41-s + 0.304·43-s + 4.52·44-s − 0.442·46-s + 0.583·47-s − 1.57·49-s + 3.32·52-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3}\)
Sign: $1$
Analytic conductor: \(318221.\)
Root analytic conductor: \(8.26269\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{8550} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((6,\ 2^{3} \cdot 3^{6} \cdot 5^{6} \cdot 19^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ 1 )\)

Particular Values

\(L(1)\) \(\approx\) \(53.08738608\)
\(L(\frac12)\) \(\approx\) \(53.08738608\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{3} \)
3 \( 1 \)
5 \( 1 \)
19$C_1$ \( ( 1 + T )^{3} \)
good7$S_4\times C_2$ \( 1 - 2 T + 15 T^{2} - 18 T^{3} + 15 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 - 5 T + 28 T^{2} - 83 T^{3} + 28 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 4 T + 31 T^{2} - 98 T^{3} + 31 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 2 T + 45 T^{2} - 58 T^{3} + 45 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + T + 24 T^{2} - 35 T^{3} + 24 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 - 5 T + 44 T^{2} - 93 T^{3} + 44 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 5 T + 72 T^{2} - 309 T^{3} + 72 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 63 T^{2} - 152 T^{3} + 63 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 - 10 T + 147 T^{2} - 824 T^{3} + 147 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - 2 T + 29 T^{2} - 334 T^{3} + 29 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 4 T + 53 T^{2} - 580 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 5 T + 86 T^{2} - 647 T^{3} + 86 p T^{4} - 5 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 12 T + 205 T^{2} - 1366 T^{3} + 205 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - T + 102 T^{2} + 153 T^{3} + 102 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 9 T + 58 T^{2} - 361 T^{3} + 58 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 16 T + 215 T^{2} - 1918 T^{3} + 215 p T^{4} - 16 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 15 T + 210 T^{2} - 1947 T^{3} + 210 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 9 T + 136 T^{2} + 713 T^{3} + 136 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 3 T + 214 T^{2} + 371 T^{3} + 214 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 9 T + 210 T^{2} - 1325 T^{3} + 210 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 6 T + 237 T^{2} + 894 T^{3} + 237 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
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

−6.67541649319250616002391150377, −6.48727385116043342871261755056, −6.41793151614214337947911746439, −6.23038905851349315809230622353, −5.90214766636485015645723240467, −5.65393143290828738959042857236, −5.48954634327354713750717670804, −5.16054866618993281672818580622, −4.95323473365551587453416534235, −4.80511196436169623808503704487, −4.30191008216256526414098040630, −4.24710223699112894916792080468, −4.17010071163398912211782834616, −3.63201615127021614725413421374, −3.61724060742348594093115607116, −3.60057328716195642293340379128, −2.75898542819860631250818462158, −2.73770053764055529537062160187, −2.73445945578938620413817865260, −1.96213978137447549832411065683, −1.87286499999235065334633400073, −1.76918623909742887775647003997, −1.01553354037126385970684511751, −0.896521199602848377152978655873, −0.75382604176035207557325065270, 0.75382604176035207557325065270, 0.896521199602848377152978655873, 1.01553354037126385970684511751, 1.76918623909742887775647003997, 1.87286499999235065334633400073, 1.96213978137447549832411065683, 2.73445945578938620413817865260, 2.73770053764055529537062160187, 2.75898542819860631250818462158, 3.60057328716195642293340379128, 3.61724060742348594093115607116, 3.63201615127021614725413421374, 4.17010071163398912211782834616, 4.24710223699112894916792080468, 4.30191008216256526414098040630, 4.80511196436169623808503704487, 4.95323473365551587453416534235, 5.16054866618993281672818580622, 5.48954634327354713750717670804, 5.65393143290828738959042857236, 5.90214766636485015645723240467, 6.23038905851349315809230622353, 6.41793151614214337947911746439, 6.48727385116043342871261755056, 6.67541649319250616002391150377

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