Properties

Label 6-4840e3-1.1-c1e3-0-3
Degree $6$
Conductor $113379904000$
Sign $-1$
Analytic cond. $57725.4$
Root an. cond. $6.21671$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·5-s − 7-s − 9-s − 2·13-s − 3·15-s − 4·17-s + 8·19-s − 21-s − 2·23-s + 6·25-s − 6·27-s − 14·29-s − 2·31-s + 3·35-s + 4·37-s − 2·39-s − 11·41-s − 29·43-s + 3·45-s + 47-s − 15·49-s − 4·51-s + 10·53-s + 8·57-s + 6·59-s + 11·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 0.377·7-s − 1/3·9-s − 0.554·13-s − 0.774·15-s − 0.970·17-s + 1.83·19-s − 0.218·21-s − 0.417·23-s + 6/5·25-s − 1.15·27-s − 2.59·29-s − 0.359·31-s + 0.507·35-s + 0.657·37-s − 0.320·39-s − 1.71·41-s − 4.42·43-s + 0.447·45-s + 0.145·47-s − 2.14·49-s − 0.560·51-s + 1.37·53-s + 1.05·57-s + 0.781·59-s + 1.40·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{9} \cdot 5^{3} \cdot 11^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{9} \cdot 5^{3} \cdot 11^{6}\)
Sign: $-1$
Analytic conductor: \(57725.4\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{9} \cdot 5^{3} \cdot 11^{6} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 + T )^{3} \)
11 \( 1 \)
good3$S_4\times C_2$ \( 1 - T + 2 T^{2} + p T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + T + 16 T^{2} + 15 T^{3} + 16 p T^{4} + p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 2 T - T^{2} - 32 T^{3} - p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 4 T + 35 T^{2} + 88 T^{3} + 35 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 8 T + 37 T^{2} - 156 T^{3} + 37 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 2 T + T^{2} - 176 T^{3} + p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 14 T + 123 T^{2} + 724 T^{3} + 123 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 2 T + 77 T^{2} + 128 T^{3} + 77 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 - 4 T + 87 T^{2} - 312 T^{3} + 87 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 11 T + 146 T^{2} + 903 T^{3} + 146 p T^{4} + 11 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 29 T + 404 T^{2} + 3343 T^{3} + 404 p T^{4} + 29 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - T + 82 T^{2} - 17 T^{3} + 82 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 - 10 T + 135 T^{2} - 1064 T^{3} + 135 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 - 6 T + 117 T^{2} - 464 T^{3} + 117 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 11 T + 154 T^{2} - 15 p T^{3} + 154 p T^{4} - 11 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 7 T + 42 T^{2} - 551 T^{3} + 42 p T^{4} - 7 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 6 T + 217 T^{2} + 840 T^{3} + 217 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 4 T - 5 T^{2} - 984 T^{3} - 5 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 6 T + 73 T^{2} + 556 T^{3} + 73 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 - 6 T + 189 T^{2} - 752 T^{3} + 189 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 13 T + 302 T^{2} + 2281 T^{3} + 302 p T^{4} + 13 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 22 T + 411 T^{2} - 4416 T^{3} + 411 p T^{4} - 22 p^{2} T^{5} + p^{3} T^{6} \)
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

−7.74697492204849863402540221024, −7.35903231802952244021587362944, −7.31806567673389138715956571654, −6.95092060606248345566448561562, −6.92832993974583204882884648032, −6.49625732992565874928720883654, −6.42675770066820725123870857226, −5.86201772621924723325634489171, −5.67863061798451843207639890762, −5.50504968734959993011773122100, −5.03331867082391896971847480087, −5.00249142760864025723263670580, −4.80204522196988233708769389595, −4.44760871120834793205685070575, −3.91925910335122155253596521094, −3.75307303940093676519528750289, −3.52133166788899808941569396554, −3.51014478496846956764418227637, −3.17260247312698151805572666843, −2.81055263729598350810259345707, −2.31324800505981573909607778969, −2.27389288075158993978811013222, −1.63814561031936494872745460438, −1.54913927904250273522743298472, −1.01550164877098077956193324648, 0, 0, 0, 1.01550164877098077956193324648, 1.54913927904250273522743298472, 1.63814561031936494872745460438, 2.27389288075158993978811013222, 2.31324800505981573909607778969, 2.81055263729598350810259345707, 3.17260247312698151805572666843, 3.51014478496846956764418227637, 3.52133166788899808941569396554, 3.75307303940093676519528750289, 3.91925910335122155253596521094, 4.44760871120834793205685070575, 4.80204522196988233708769389595, 5.00249142760864025723263670580, 5.03331867082391896971847480087, 5.50504968734959993011773122100, 5.67863061798451843207639890762, 5.86201772621924723325634489171, 6.42675770066820725123870857226, 6.49625732992565874928720883654, 6.92832993974583204882884648032, 6.95092060606248345566448561562, 7.31806567673389138715956571654, 7.35903231802952244021587362944, 7.74697492204849863402540221024

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