↑

Properties

Label 6-9360e3-1.1-c1e3-0-6
Degree $6$
Conductor $820025856000$
Sign $-1$
Analytic cond. $417502.$
Root an. cond. $8.64522$
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  + 3·5-s − 3·7-s − 3·11-s + 3·13-s + 3·17-s − 6·19-s − 3·23-s + 6·25-s − 6·31-s − 9·35-s + 3·37-s − 3·41-s − 6·43-s − 6·47-s − 6·49-s − 3·53-s − 9·55-s + 9·61-s + 9·65-s − 12·67-s − 3·71-s + 9·77-s − 15·79-s + 9·85-s − 9·89-s − 9·91-s − 18·95-s + ⋯
L(s)  = 1  + 1.34·5-s − 1.13·7-s − 0.904·11-s + 0.832·13-s + 0.727·17-s − 1.37·19-s − 0.625·23-s + 6/5·25-s − 1.07·31-s − 1.52·35-s + 0.493·37-s − 0.468·41-s − 0.914·43-s − 0.875·47-s − 6/7·49-s − 0.412·53-s − 1.21·55-s + 1.15·61-s + 1.11·65-s − 1.46·67-s − 0.356·71-s + 1.02·77-s − 1.68·79-s + 0.976·85-s − 0.953·89-s − 0.943·91-s − 1.84·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3}\)
Sign: $-1$
Analytic conductor: \(417502.\)
Root analytic conductor: \(8.64522\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{12} \cdot 3^{6} \cdot 5^{3} \cdot 13^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
13$C_1$ \( ( 1 - T )^{3} \)
good7$S_4\times C_2$ \( 1 + 3 T + 15 T^{2} + 30 T^{3} + 15 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 9 T^{2} + 2 T^{3} + 9 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 - 3 T + 45 T^{2} - 98 T^{3} + 45 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 6 T + 45 T^{2} + 212 T^{3} + 45 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 3 T + 63 T^{2} + 134 T^{3} + 63 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 63 T^{2} + 24 T^{3} + 63 p T^{4} + p^{3} T^{6} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{3} \)
37$S_4\times C_2$ \( 1 - 3 T + 75 T^{2} - 258 T^{3} + 75 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 3 T + 39 T^{2} + 442 T^{3} + 39 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 6 T + 105 T^{2} + 420 T^{3} + 105 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 6 T + 117 T^{2} + 532 T^{3} + 117 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 3 T + 135 T^{2} + 254 T^{3} + 135 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 81 T^{2} + 192 T^{3} + 81 p T^{4} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 9 T + 171 T^{2} - 934 T^{3} + 171 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 12 T + 213 T^{2} + 1560 T^{3} + 213 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 3 T + 93 T^{2} + 730 T^{3} + 93 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 147 T^{2} - 232 T^{3} + 147 p T^{4} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 15 T + 225 T^{2} + 1778 T^{3} + 225 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
83$C_2$ \( ( 1 + p T^{2} )^{3} \)
89$S_4\times C_2$ \( 1 + 9 T + 207 T^{2} + 1086 T^{3} + 207 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 15 T + 225 T^{2} - 2282 T^{3} + 225 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
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

−7.15409905737285861406364117489, −6.64610207593955251115850847991, −6.63145355071895538658633299444, −6.60961608012859868231133660310, −6.09692716335981951469214933149, −6.07394020076234699469849713974, −5.89633975309209762648521162221, −5.47977400700478075115556361339, −5.41370317076343167388292944781, −5.19685479392636248902449802718, −4.87040668502057591456651784425, −4.62312739698384960778813438083, −4.37996510412988060958940479604, −3.91432385897926831202811014653, −3.78105204179650240276122487077, −3.70824730599394133829152529529, −3.09304587986748031461467579496, −2.97629336444665369251372725514, −2.94492351034668089122890874408, −2.41672462203813997850950356886, −2.12198875666129722741725897766, −2.06684480650517307205142538409, −1.37968224708999976594693607659, −1.35834798959458669168368352894, −1.13921539565682475689184121140, 0, 0, 0, 1.13921539565682475689184121140, 1.35834798959458669168368352894, 1.37968224708999976594693607659, 2.06684480650517307205142538409, 2.12198875666129722741725897766, 2.41672462203813997850950356886, 2.94492351034668089122890874408, 2.97629336444665369251372725514, 3.09304587986748031461467579496, 3.70824730599394133829152529529, 3.78105204179650240276122487077, 3.91432385897926831202811014653, 4.37996510412988060958940479604, 4.62312739698384960778813438083, 4.87040668502057591456651784425, 5.19685479392636248902449802718, 5.41370317076343167388292944781, 5.47977400700478075115556361339, 5.89633975309209762648521162221, 6.07394020076234699469849713974, 6.09692716335981951469214933149, 6.60961608012859868231133660310, 6.63145355071895538658633299444, 6.64610207593955251115850847991, 7.15409905737285861406364117489

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