Properties

Label 6-7360e3-1.1-c1e3-0-9
Degree $6$
Conductor $398688256000$
Sign $-1$
Analytic cond. $202985.$
Root an. cond. $7.66615$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $3$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 3·5-s + 3·7-s − 3·9-s − 3·11-s − 3·17-s − 3·19-s + 3·23-s + 6·25-s − 3·27-s − 3·29-s + 6·31-s + 9·35-s − 12·37-s − 6·41-s − 18·43-s − 9·45-s + 15·47-s − 9·49-s − 6·53-s − 9·55-s − 6·59-s − 27·61-s − 9·63-s − 12·67-s + 6·71-s − 15·73-s − 9·77-s + ⋯
L(s)  = 1  + 1.34·5-s + 1.13·7-s − 9-s − 0.904·11-s − 0.727·17-s − 0.688·19-s + 0.625·23-s + 6/5·25-s − 0.577·27-s − 0.557·29-s + 1.07·31-s + 1.52·35-s − 1.97·37-s − 0.937·41-s − 2.74·43-s − 1.34·45-s + 2.18·47-s − 9/7·49-s − 0.824·53-s − 1.21·55-s − 0.781·59-s − 3.45·61-s − 1.13·63-s − 1.46·67-s + 0.712·71-s − 1.75·73-s − 1.02·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{3} \cdot 23^{3}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(6\)
Conductor: \(2^{18} \cdot 5^{3} \cdot 23^{3}\)
Sign: $-1$
Analytic conductor: \(202985.\)
Root analytic conductor: \(7.66615\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(3\)
Selberg data: \((6,\ 2^{18} \cdot 5^{3} \cdot 23^{3} ,\ ( \ : 1/2, 1/2, 1/2 ),\ -1 )\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{3} \)
23$C_1$ \( ( 1 - T )^{3} \)
good3$S_4\times C_2$ \( 1 + p T^{2} + p T^{3} + p^{2} T^{4} + p^{3} T^{6} \)
7$S_4\times C_2$ \( 1 - 3 T + 18 T^{2} - 40 T^{3} + 18 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} \)
11$S_4\times C_2$ \( 1 + 3 T + 18 T^{2} + 78 T^{3} + 18 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
13$S_4\times C_2$ \( 1 + 21 T^{2} - 29 T^{3} + 21 p T^{4} + p^{3} T^{6} \)
17$S_4\times C_2$ \( 1 + 3 T + 36 T^{2} + 114 T^{3} + 36 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
19$S_4\times C_2$ \( 1 + 3 T + 24 T^{2} + 162 T^{3} + 24 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
29$S_4\times C_2$ \( 1 + 3 T + 81 T^{2} + 162 T^{3} + 81 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} \)
31$S_4\times C_2$ \( 1 - 6 T + 51 T^{2} - 123 T^{3} + 51 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{3} \)
41$S_4\times C_2$ \( 1 + 6 T + 63 T^{2} + 423 T^{3} + 63 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
43$S_4\times C_2$ \( 1 + 18 T + 201 T^{2} + 1516 T^{3} + 201 p T^{4} + 18 p^{2} T^{5} + p^{3} T^{6} \)
47$S_4\times C_2$ \( 1 - 15 T + 207 T^{2} - 1486 T^{3} + 207 p T^{4} - 15 p^{2} T^{5} + p^{3} T^{6} \)
53$S_4\times C_2$ \( 1 + 6 T + 27 T^{2} + 100 T^{3} + 27 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
59$S_4\times C_2$ \( 1 + 6 T + 141 T^{2} + 676 T^{3} + 141 p T^{4} + 6 p^{2} T^{5} + p^{3} T^{6} \)
61$S_4\times C_2$ \( 1 + 27 T + 408 T^{2} + 3890 T^{3} + 408 p T^{4} + 27 p^{2} T^{5} + p^{3} T^{6} \)
67$S_4\times C_2$ \( 1 + 12 T - 39 T^{2} - 1336 T^{3} - 39 p T^{4} + 12 p^{2} T^{5} + p^{3} T^{6} \)
71$S_4\times C_2$ \( 1 - 6 T + 153 T^{2} - 649 T^{3} + 153 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} \)
73$S_4\times C_2$ \( 1 + 15 T + 177 T^{2} + 1602 T^{3} + 177 p T^{4} + 15 p^{2} T^{5} + p^{3} T^{6} \)
79$C_2$ \( ( 1 + p T^{2} )^{3} \)
83$S_4\times C_2$ \( 1 - 12 T + 189 T^{2} - 1928 T^{3} + 189 p T^{4} - 12 p^{2} T^{5} + p^{3} T^{6} \)
89$S_4\times C_2$ \( 1 + 30 T + 531 T^{2} + 5948 T^{3} + 531 p T^{4} + 30 p^{2} T^{5} + p^{3} T^{6} \)
97$S_4\times C_2$ \( 1 - 9 T + 222 T^{2} - 1608 T^{3} + 222 p T^{4} - 9 p^{2} T^{5} + p^{3} T^{6} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.30723732807434238596446564445, −6.89730674169682433157466639353, −6.88562183345358459093978748305, −6.73318037720067358348389536062, −6.25493707410739750491931949503, −6.22143592589883443256167128761, −5.95289752306703574529671417942, −5.50307234965556532260995459038, −5.42247049286821741479498815905, −5.39071433512401648978861199375, −4.96532871717609685616097072926, −4.70511953522468432430166635787, −4.67613924898852227529903444295, −4.31652677321100162199616682314, −4.00929995949813386172178021753, −3.65613653552087725819117838273, −3.18076753213363444956887035080, −3.05275107546769580532327697328, −2.91095849709243152740161292383, −2.42375052368160754196744215254, −2.40829685258919769041652134691, −1.82519762556612294696473167030, −1.58974213402113363185029212796, −1.39088886790977214020708697577, −1.31426091749373818876591532900, 0, 0, 0, 1.31426091749373818876591532900, 1.39088886790977214020708697577, 1.58974213402113363185029212796, 1.82519762556612294696473167030, 2.40829685258919769041652134691, 2.42375052368160754196744215254, 2.91095849709243152740161292383, 3.05275107546769580532327697328, 3.18076753213363444956887035080, 3.65613653552087725819117838273, 4.00929995949813386172178021753, 4.31652677321100162199616682314, 4.67613924898852227529903444295, 4.70511953522468432430166635787, 4.96532871717609685616097072926, 5.39071433512401648978861199375, 5.42247049286821741479498815905, 5.50307234965556532260995459038, 5.95289752306703574529671417942, 6.22143592589883443256167128761, 6.25493707410739750491931949503, 6.73318037720067358348389536062, 6.88562183345358459093978748305, 6.89730674169682433157466639353, 7.30723732807434238596446564445

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