Properties

Label 4-54000-1.1-c1e2-0-4
Degree $4$
Conductor $54000$
Sign $-1$
Analytic cond. $3.44308$
Root an. cond. $1.36218$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s + 5-s + 6-s + 3·8-s + 9-s − 10-s − 8·11-s + 12-s − 15-s − 16-s + 4·17-s − 18-s − 20-s + 8·22-s − 3·24-s + 25-s − 27-s + 30-s − 5·32-s + 8·33-s − 4·34-s − 36-s + 3·40-s + 8·43-s + 8·44-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s − 2.41·11-s + 0.288·12-s − 0.258·15-s − 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.223·20-s + 1.70·22-s − 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.182·30-s − 0.883·32-s + 1.39·33-s − 0.685·34-s − 1/6·36-s + 0.474·40-s + 1.21·43-s + 1.20·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(54000\)    =    \(2^{4} \cdot 3^{3} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(3.44308\)
Root analytic conductor: \(1.36218\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{54000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 54000,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + T + p T^{2} \)
3$C_1$ \( 1 + T \)
5$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.760887039888426379121330275173, −9.523451675812284265792380668213, −8.772712605956389004925762991206, −8.132479360823444390461948599740, −7.66488013441745380243230842523, −7.62818581639541327480210566204, −6.59119797171035359138537025559, −6.00095562582224088382030415028, −5.23920392624592057055772361749, −5.10854764303702027975036605471, −4.42469036791545275362320561796, −3.36550561270118716838366308509, −2.59055487455418232206827677965, −1.46780416341541155024498465553, 0, 1.46780416341541155024498465553, 2.59055487455418232206827677965, 3.36550561270118716838366308509, 4.42469036791545275362320561796, 5.10854764303702027975036605471, 5.23920392624592057055772361749, 6.00095562582224088382030415028, 6.59119797171035359138537025559, 7.62818581639541327480210566204, 7.66488013441745380243230842523, 8.132479360823444390461948599740, 8.772712605956389004925762991206, 9.523451675812284265792380668213, 9.760887039888426379121330275173

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