Properties

Label 4-3120e2-1.1-c1e2-0-12
Degree $4$
Conductor $9734400$
Sign $1$
Analytic cond. $620.673$
Root an. cond. $4.99132$
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  + 4·5-s − 9-s − 4·11-s + 4·19-s + 11·25-s + 4·29-s + 8·31-s − 12·41-s − 4·45-s − 2·49-s − 16·55-s − 28·59-s + 20·61-s − 16·71-s − 16·79-s + 81-s + 36·89-s + 16·95-s + 4·99-s − 28·101-s − 24·109-s − 10·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 1.20·11-s + 0.917·19-s + 11/5·25-s + 0.742·29-s + 1.43·31-s − 1.87·41-s − 0.596·45-s − 2/7·49-s − 2.15·55-s − 3.64·59-s + 2.56·61-s − 1.89·71-s − 1.80·79-s + 1/9·81-s + 3.81·89-s + 1.64·95-s + 0.402·99-s − 2.78·101-s − 2.29·109-s − 0.909·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(9734400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(620.673\)
Root analytic conductor: \(4.99132\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3120} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 9734400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.020388928\)
\(L(\frac12)\) \(\approx\) \(3.020388928\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
13$C_2$ \( 1 + T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 38 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 46 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−9.158093684458620614520832841026, −8.483156927793049768175057895261, −8.203396142097353702658192561100, −7.73684288564244224274223851958, −7.51461354384254419417809268475, −6.72837551171062963927816350481, −6.62548920817660021423464808585, −6.31509713445386782909366190284, −5.72211124218642380475171809840, −5.39054375769800099634186200050, −5.29036757156116844128161869765, −4.58438088874882496699477259999, −4.56153144322455286771614073684, −3.59120485750435785431552244293, −3.05962880074096077920680276795, −2.75464441049776245883667430030, −2.49223479740000434515089261364, −1.63632091923718299215314129535, −1.46243885141435954861040462657, −0.52558357502861835195655003683, 0.52558357502861835195655003683, 1.46243885141435954861040462657, 1.63632091923718299215314129535, 2.49223479740000434515089261364, 2.75464441049776245883667430030, 3.05962880074096077920680276795, 3.59120485750435785431552244293, 4.56153144322455286771614073684, 4.58438088874882496699477259999, 5.29036757156116844128161869765, 5.39054375769800099634186200050, 5.72211124218642380475171809840, 6.31509713445386782909366190284, 6.62548920817660021423464808585, 6.72837551171062963927816350481, 7.51461354384254419417809268475, 7.73684288564244224274223851958, 8.203396142097353702658192561100, 8.483156927793049768175057895261, 9.158093684458620614520832841026

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