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Properties

Label 4-1200e2-1.1-c0e2-0-3
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  + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s − 2·17-s + 2·19-s + 2·23-s + 4·27-s − 3·36-s − 2·47-s + 2·48-s − 4·51-s − 4·53-s + 4·57-s − 2·61-s − 64-s + 2·68-s + 4·69-s − 2·76-s + 5·81-s − 2·92-s − 4·108-s − 2·109-s − 2·113-s + 127-s + 131-s + ⋯
L(s)  = 1  + 2·3-s − 4-s + 3·9-s − 2·12-s + 16-s − 2·17-s + 2·19-s + 2·23-s + 4·27-s − 3·36-s − 2·47-s + 2·48-s − 4·51-s − 4·53-s + 4·57-s − 2·61-s − 64-s + 2·68-s + 4·69-s − 2·76-s + 5·81-s − 2·92-s − 4·108-s − 2·109-s − 2·113-s + 127-s + 131-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

\(L(\frac{1}{2})\) \(\approx\) \(1.824231506\)
\(L(\frac12)\) \(\approx\) \(1.824231506\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
3$C_1$ \( ( 1 - T )^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 + T^{4} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_2$ \( ( 1 + T^{2} )^{2} \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_1$ \( ( 1 + T )^{4} \)
59$C_2^2$ \( 1 + T^{4} \)
61$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
67$C_2$ \( ( 1 + T^{2} )^{2} \)
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$C_2$ \( ( 1 + T^{2} )^{2} \)
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

−9.637450906837677117328221248887, −9.552912523416957621137522193713, −9.300539497930004516676687139493, −9.143732979727727472447007248396, −8.473306196265106933400171872393, −8.247735040832911389963267419658, −7.83577268762492082612155695589, −7.47034110889831128408493898925, −6.95283442066884362235293910954, −6.63549712264577725856802319743, −6.10109183576918176047941115420, −5.05842061467781748152997708081, −4.96134257722837535038927549667, −4.54017112006405775258489095369, −4.04498607870832072719601098739, −3.29278018354920551342566757020, −3.13171738102636819442293293872, −2.73250934799959549399788528511, −1.69184890118543220032359375778, −1.33429735571287586165829232487, 1.33429735571287586165829232487, 1.69184890118543220032359375778, 2.73250934799959549399788528511, 3.13171738102636819442293293872, 3.29278018354920551342566757020, 4.04498607870832072719601098739, 4.54017112006405775258489095369, 4.96134257722837535038927549667, 5.05842061467781748152997708081, 6.10109183576918176047941115420, 6.63549712264577725856802319743, 6.95283442066884362235293910954, 7.47034110889831128408493898925, 7.83577268762492082612155695589, 8.247735040832911389963267419658, 8.473306196265106933400171872393, 9.143732979727727472447007248396, 9.300539497930004516676687139493, 9.552912523416957621137522193713, 9.637450906837677117328221248887

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