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Properties

Label 4-1200e2-1.1-c0e2-0-0
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $0.358654$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 9-s + 5·16-s − 2·17-s + 2·18-s − 2·19-s + 2·23-s − 6·32-s + 4·34-s − 3·36-s + 4·38-s − 4·46-s + 2·47-s − 2·61-s + 7·64-s − 6·68-s + 4·72-s − 6·76-s + 81-s + 6·92-s − 4·94-s + 2·109-s − 2·113-s + 4·122-s + 127-s + ⋯
L(s)  = 1  − 2·2-s + 3·4-s − 4·8-s − 9-s + 5·16-s − 2·17-s + 2·18-s − 2·19-s + 2·23-s − 6·32-s + 4·34-s − 3·36-s + 4·38-s − 4·46-s + 2·47-s − 2·61-s + 7·64-s − 6·68-s + 4·72-s − 6·76-s + 81-s + 6·92-s − 4·94-s + 2·109-s − 2·113-s + 4·122-s + 127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.358654\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2827514228\)
\(L(\frac12)\) \(\approx\) \(0.2827514228\)
\(L(1)\) 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$C_1$ \( ( 1 + T )^{2} \)
3$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
29$C_2^2$ \( 1 + T^{4} \)
31$C_2$ \( ( 1 + T^{2} )^{2} \)
37$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−10.33627604212210333964710070170, −9.348672313368954692765408602787, −9.251065716201073465975765640428, −8.958819518630306750735421541738, −8.708829157383072054403462914759, −8.162484288652240273544613677108, −8.016274656817931161894764196798, −7.22514497544766569979336867921, −6.97856812361007893798053660801, −6.63894696983814711104810248890, −6.15953677823768216065772613776, −5.86256817537407808931962074365, −5.23600874176043722946185523744, −4.53725946997020161386281139161, −4.02476316341054175305399621095, −3.08946936508653771795112343695, −2.80805086292252744111457222456, −2.19716204253244696663132773602, −1.78190024764347450669228785225, −0.61960754735892546299978320216, 0.61960754735892546299978320216, 1.78190024764347450669228785225, 2.19716204253244696663132773602, 2.80805086292252744111457222456, 3.08946936508653771795112343695, 4.02476316341054175305399621095, 4.53725946997020161386281139161, 5.23600874176043722946185523744, 5.86256817537407808931962074365, 6.15953677823768216065772613776, 6.63894696983814711104810248890, 6.97856812361007893798053660801, 7.22514497544766569979336867921, 8.016274656817931161894764196798, 8.162484288652240273544613677108, 8.708829157383072054403462914759, 8.958819518630306750735421541738, 9.251065716201073465975765640428, 9.348672313368954692765408602787, 10.33627604212210333964710070170

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