Properties

Label 4-108000-1.1-c1e2-0-1
Degree $4$
Conductor $108000$
Sign $-1$
Analytic cond. $6.88617$
Root an. cond. $1.61992$
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 − 8·7-s − 8-s + 9-s + 10-s + 12-s + 8·14-s − 15-s + 16-s + 12·17-s − 18-s − 20-s − 8·21-s − 24-s + 25-s + 27-s − 8·28-s + 30-s − 32-s − 12·34-s + 8·35-s + 36-s + 40-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 3.02·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s + 2.13·14-s − 0.258·15-s + 1/4·16-s + 2.91·17-s − 0.235·18-s − 0.223·20-s − 1.74·21-s − 0.204·24-s + 1/5·25-s + 0.192·27-s − 1.51·28-s + 0.182·30-s − 0.176·32-s − 2.05·34-s + 1.35·35-s + 1/6·36-s + 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(108000\)    =    \(2^{5} \cdot 3^{3} \cdot 5^{3}\)
Sign: $-1$
Analytic conductor: \(6.88617\)
Root analytic conductor: \(1.61992\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{108000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 108000,\ (\ :1/2, 1/2),\ -1)\)

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$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
5$C_1$ \( 1 + T \)
good7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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

−9.305587122869086271308905422277, −9.116818239719236618689703406908, −8.101485244373939343160290559425, −7.941231322257043910223245152113, −7.35042654382863533673550380313, −6.78354731906038205235867209347, −6.42217617652666865799421983967, −5.87146842228725166509134997887, −5.34762325653913158633333815914, −4.22720411424127134624171462120, −3.36585804145949552210873743484, −3.25067910161980170709704604527, −2.84728890251355077691674063068, −1.34557356174820418480074519421, 0, 1.34557356174820418480074519421, 2.84728890251355077691674063068, 3.25067910161980170709704604527, 3.36585804145949552210873743484, 4.22720411424127134624171462120, 5.34762325653913158633333815914, 5.87146842228725166509134997887, 6.42217617652666865799421983967, 6.78354731906038205235867209347, 7.35042654382863533673550380313, 7.941231322257043910223245152113, 8.101485244373939343160290559425, 9.116818239719236618689703406908, 9.305587122869086271308905422277

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