Properties

Label 4-1440e2-1.1-c1e2-0-8
Degree $4$
Conductor $2073600$
Sign $1$
Analytic cond. $132.214$
Root an. cond. $3.39093$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 16·19-s − 25-s − 12·29-s − 16·31-s − 12·41-s − 2·49-s − 12·61-s + 32·71-s + 16·79-s − 20·89-s − 32·95-s − 28·101-s + 20·109-s − 22·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 24·145-s + 149-s + 151-s + 32·155-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  − 0.894·5-s + 3.67·19-s − 1/5·25-s − 2.22·29-s − 2.87·31-s − 1.87·41-s − 2/7·49-s − 1.53·61-s + 3.79·71-s + 1.80·79-s − 2.11·89-s − 3.28·95-s − 2.78·101-s + 1.91·109-s − 2·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.99·145-s + 0.0819·149-s + 0.0813·151-s + 2.57·155-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2073600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2073600\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(132.214\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1440} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2073600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.212335175\)
\(L(\frac12)\) \(\approx\) \(1.212335175\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3 \( 1 \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−9.610342966639099694925149433255, −9.333477879231934759220167449011, −9.226058171897001814659227291784, −8.265117223089157239411512886960, −8.209151443680084889922028916982, −7.52342135744687534022378169885, −7.41144368419041654986731398540, −7.13402074126162639589090427692, −6.64504622952495083972104400827, −5.79360653219560157313105291554, −5.57248287375360996119390763337, −5.10987422315487353294961914387, −4.96621721210875437351677184952, −3.85664316549330218630053728810, −3.71599304262311115499967452216, −3.41851728508801929838304482548, −2.85350458962345062408189968866, −1.87322452308456859788832528505, −1.49057005351059371121255204327, −0.45336944881666730517742830952, 0.45336944881666730517742830952, 1.49057005351059371121255204327, 1.87322452308456859788832528505, 2.85350458962345062408189968866, 3.41851728508801929838304482548, 3.71599304262311115499967452216, 3.85664316549330218630053728810, 4.96621721210875437351677184952, 5.10987422315487353294961914387, 5.57248287375360996119390763337, 5.79360653219560157313105291554, 6.64504622952495083972104400827, 7.13402074126162639589090427692, 7.41144368419041654986731398540, 7.52342135744687534022378169885, 8.209151443680084889922028916982, 8.265117223089157239411512886960, 9.226058171897001814659227291784, 9.333477879231934759220167449011, 9.610342966639099694925149433255

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