Properties

Label 4-54000-1.1-c1e2-0-5
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

−9.965958896202437987484594622667, −8.982199030019329801304860528684, −8.843956595114680827452383410006, −8.270140448885357284476919824336, −7.68003844385765985083943135134, −7.40264879408133961073539513214, −6.55157891837643531715592264808, −5.86497653842965752491328535183, −5.32469243442401148699352691061, −4.54153907278640853760279816056, −4.52160444884874669934496061646, −3.29035819756263414840801407793, −3.01549784140686353642765162463, −2.11935007719153584295308930473, 0, 2.11935007719153584295308930473, 3.01549784140686353642765162463, 3.29035819756263414840801407793, 4.52160444884874669934496061646, 4.54153907278640853760279816056, 5.32469243442401148699352691061, 5.86497653842965752491328535183, 6.55157891837643531715592264808, 7.40264879408133961073539513214, 7.68003844385765985083943135134, 8.270140448885357284476919824336, 8.843956595114680827452383410006, 8.982199030019329801304860528684, 9.965958896202437987484594622667

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