Properties

Label 6-4840e3-1.1-c1e3-0-4
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 − 3·7-s − 9-s + 2·13-s − 3·15-s − 4·17-s + 4·19-s − 3·21-s − 6·23-s + 6·25-s + 6·27-s − 2·29-s − 22·31-s + 9·35-s − 4·37-s + 2·39-s − 5·41-s + 43-s + 3·45-s − 7·47-s + 5·49-s − 4·51-s − 6·53-s + 4·57-s − 6·59-s + 21·61-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.34·5-s − 1.13·7-s − 1/3·9-s + 0.554·13-s − 0.774·15-s − 0.970·17-s + 0.917·19-s − 0.654·21-s − 1.25·23-s + 6/5·25-s + 1.15·27-s − 0.371·29-s − 3.95·31-s + 1.52·35-s − 0.657·37-s + 0.320·39-s − 0.780·41-s + 0.152·43-s + 0.447·45-s − 1.02·47-s + 5/7·49-s − 0.560·51-s − 0.824·53-s + 0.529·57-s − 0.781·59-s + 2.68·61-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$S_4\times C_2$ \( 1 - T + 2 T^{2} - p^{2} T^{3} + 2 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 + 3 T + 4 T^{2} - 11 T^{3} + 4 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 - 2 T + 31 T^{2} - 48 T^{3} + 31 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \)
17$D_{6}$ \( 1 + 4 T + 3 T^{2} - 8 T^{3} + 3 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 - 4 T + 53 T^{2} - 140 T^{3} + 53 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
23$S_4\times C_2$ \( 1 + 6 T + 49 T^{2} + 264 T^{3} + 49 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 2 T + 59 T^{2} + 140 T^{3} + 59 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 + 22 T + 245 T^{2} + 1688 T^{3} + 245 p T^{4} + 22 p^{2} T^{5} + p^{3} T^{6} \)
37$S_4\times C_2$ \( 1 + 4 T + 87 T^{2} + 216 T^{3} + 87 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} \)
41$S_4\times C_2$ \( 1 + 5 T + 122 T^{2} + 401 T^{3} + 122 p T^{4} + 5 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 - T + 48 T^{2} + 189 T^{3} + 48 p T^{4} - p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 + 7 T + 106 T^{2} + 667 T^{3} + 106 p T^{4} + 7 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 87 T^{2} + 8 p T^{3} + 87 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 157 T^{2} + 696 T^{3} + 157 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 - 21 T + 298 T^{2} - 2637 T^{3} + 298 p T^{4} - 21 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 - 15 T + 210 T^{2} - 1659 T^{3} + 210 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 + 10 T + 17 T^{2} - 576 T^{3} + 17 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 - 4 T + 187 T^{2} - 552 T^{3} + 187 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} \)
79$S_4\times C_2$ \( 1 + 2 T + 185 T^{2} + 356 T^{3} + 185 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
83$S_4\times C_2$ \( 1 + 10 T + 237 T^{2} + 1480 T^{3} + 237 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 - 19 T + 334 T^{2} - 3223 T^{3} + 334 p T^{4} - 19 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 + 2 T + 283 T^{2} + 384 T^{3} + 283 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} \)
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

−7.78426710909296396077531547271, −7.38766459669942561281154620972, −7.26656378001658168549144649870, −6.90883020482644245739039379353, −6.71141433712892672640809671801, −6.61867524070185091112823373460, −6.44310483888822543422705977202, −5.95148742667249809249055349404, −5.60014136939970169714439414644, −5.50115034486687796686463433972, −5.28945004548477289677995995449, −5.01294993360133391370697453226, −4.50887789449460731836108781181, −4.47673172466023877623638578164, −3.90131488884499232120354911703, −3.81189834880792461620183178756, −3.47481012678752668610996228637, −3.43547284125106784883258201782, −3.34986226032756410216320173333, −2.63310381922910535886419958531, −2.51813604807878510736711395143, −2.22207871065707618178622017280, −1.80376588206215101272348193451, −1.22955211254888184714218590206, −1.13510304525923839941684458998, 0, 0, 0, 1.13510304525923839941684458998, 1.22955211254888184714218590206, 1.80376588206215101272348193451, 2.22207871065707618178622017280, 2.51813604807878510736711395143, 2.63310381922910535886419958531, 3.34986226032756410216320173333, 3.43547284125106784883258201782, 3.47481012678752668610996228637, 3.81189834880792461620183178756, 3.90131488884499232120354911703, 4.47673172466023877623638578164, 4.50887789449460731836108781181, 5.01294993360133391370697453226, 5.28945004548477289677995995449, 5.50115034486687796686463433972, 5.60014136939970169714439414644, 5.95148742667249809249055349404, 6.44310483888822543422705977202, 6.61867524070185091112823373460, 6.71141433712892672640809671801, 6.90883020482644245739039379353, 7.26656378001658168549144649870, 7.38766459669942561281154620972, 7.78426710909296396077531547271

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