Properties

Label 4-1200e2-1.1-c2e2-0-2
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $1069.13$
Root an. cond. $5.71818$
Motivic weight $2$
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  − 3-s − 16·7-s − 8·9-s + 4·13-s − 22·19-s + 16·21-s + 17·27-s + 92·31-s − 32·37-s − 4·39-s − 124·43-s + 94·49-s + 22·57-s − 32·61-s + 128·63-s − 226·67-s + 202·73-s − 136·79-s + 55·81-s − 64·91-s − 92·93-s − 44·97-s − 52·103-s + 352·109-s + 32·111-s − 32·117-s − 73·121-s + ⋯
L(s)  = 1  − 1/3·3-s − 2.28·7-s − 8/9·9-s + 4/13·13-s − 1.15·19-s + 0.761·21-s + 0.629·27-s + 2.96·31-s − 0.864·37-s − 0.102·39-s − 2.88·43-s + 1.91·49-s + 0.385·57-s − 0.524·61-s + 2.03·63-s − 3.37·67-s + 2.76·73-s − 1.72·79-s + 0.679·81-s − 0.703·91-s − 0.989·93-s − 0.453·97-s − 0.504·103-s + 3.22·109-s + 0.288·111-s − 0.273·117-s − 0.603·121-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1069.13\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.01349444821\)
\(L(\frac12)\) \(\approx\) \(0.01349444821\)
\(L(2)\) 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$C_2$ \( 1 + T + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2$ \( ( 1 + 8 T + p^{2} T^{2} )^{2} \)
11$C_2$ \( ( 1 - 13 T + p^{2} T^{2} )( 1 + 13 T + p^{2} T^{2} ) \)
13$C_2$ \( ( 1 - 2 T + p^{2} T^{2} )^{2} \)
17$C_2$ \( ( 1 - 29 T + p^{2} T^{2} )( 1 + 29 T + p^{2} T^{2} ) \)
19$C_2$ \( ( 1 + 11 T + p^{2} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 202 T^{2} + p^{4} T^{4} \)
29$C_2^2$ \( 1 - 422 T^{2} + p^{4} T^{4} \)
31$C_2$ \( ( 1 - 46 T + p^{2} T^{2} )^{2} \)
37$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
41$C_2^2$ \( 1 - 527 T^{2} + p^{4} T^{4} \)
43$C_2$ \( ( 1 + 62 T + p^{2} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 3158 T^{2} + p^{4} T^{4} \)
53$C_2^2$ \( 1 - 4358 T^{2} + p^{4} T^{4} \)
59$C_2^2$ \( 1 - 1922 T^{2} + p^{4} T^{4} \)
61$C_2$ \( ( 1 + 16 T + p^{2} T^{2} )^{2} \)
67$C_2$ \( ( 1 + 113 T + p^{2} T^{2} )^{2} \)
71$C_2^2$ \( 1 + 1258 T^{2} + p^{4} T^{4} \)
73$C_2$ \( ( 1 - 101 T + p^{2} T^{2} )^{2} \)
79$C_2$ \( ( 1 + 68 T + p^{2} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 13463 T^{2} + p^{4} T^{4} \)
89$C_2^2$ \( 1 - 13007 T^{2} + p^{4} T^{4} \)
97$C_2$ \( ( 1 + 22 T + p^{2} T^{2} )^{2} \)
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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

−10.01274096032803513591751294553, −9.605289258876036148875300587766, −8.731324061530895798557912495062, −8.623996292689705359198859012905, −8.380367797583675783034876398206, −7.72742869616064280849844850322, −7.08091211860299870299219031553, −6.64354968376629565416730529172, −6.45731926755107943914860482143, −6.00103506612536207990726141795, −5.91902057545992538086305392346, −4.92836355218241569973819037582, −4.76513552661245526947322344817, −4.06346019821561281276721173990, −3.29830378084458742007142922255, −3.22668108765024435143525169275, −2.72367440186264164802364577018, −1.99474838729571371493396398399, −1.03259328635969993888585228875, −0.03991944064517858759358523238, 0.03991944064517858759358523238, 1.03259328635969993888585228875, 1.99474838729571371493396398399, 2.72367440186264164802364577018, 3.22668108765024435143525169275, 3.29830378084458742007142922255, 4.06346019821561281276721173990, 4.76513552661245526947322344817, 4.92836355218241569973819037582, 5.91902057545992538086305392346, 6.00103506612536207990726141795, 6.45731926755107943914860482143, 6.64354968376629565416730529172, 7.08091211860299870299219031553, 7.72742869616064280849844850322, 8.380367797583675783034876398206, 8.623996292689705359198859012905, 8.731324061530895798557912495062, 9.605289258876036148875300587766, 10.01274096032803513591751294553

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