Properties

Label 4-30e4-1.1-c5e2-0-15
Degree $4$
Conductor $810000$
Sign $1$
Analytic cond. $20835.6$
Root an. cond. $12.0143$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 22·7-s − 372·11-s − 10·13-s + 1.21e3·17-s − 1.55e3·19-s + 2.65e3·23-s + 1.81e3·29-s + 1.09e4·31-s − 3.31e3·37-s − 1.90e4·41-s − 1.22e4·43-s − 1.52e4·47-s − 8.05e3·49-s + 2.07e4·53-s − 1.87e4·59-s + 3.57e4·61-s + 9.81e4·67-s − 1.14e5·71-s − 1.09e5·73-s + 8.18e3·77-s + 9.55e4·79-s + 3.73e3·83-s − 9.36e4·89-s + 220·91-s + 9.18e4·97-s − 2.32e5·101-s − 5.12e4·103-s + ⋯
L(s)  = 1  − 0.169·7-s − 0.926·11-s − 0.0164·13-s + 1.01·17-s − 0.985·19-s + 1.04·23-s + 0.400·29-s + 2.04·31-s − 0.398·37-s − 1.77·41-s − 1.01·43-s − 1.00·47-s − 0.479·49-s + 1.01·53-s − 0.699·59-s + 1.22·61-s + 2.67·67-s − 2.68·71-s − 2.41·73-s + 0.157·77-s + 1.72·79-s + 0.0594·83-s − 1.25·89-s + 0.00278·91-s + 0.991·97-s − 2.26·101-s − 0.475·103-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(20835.6\)
Root analytic conductor: \(12.0143\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 810000,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 \)
good7$D_{4}$ \( 1 + 22 T + 8535 T^{2} + 22 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 + 372 T + 129898 T^{2} + 372 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 10 T + 339411 T^{2} + 10 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 1212 T + 2980150 T^{2} - 1212 p^{5} T^{3} + p^{10} T^{4} \)
19$D_{4}$ \( 1 + 1550 T + 2503623 T^{2} + 1550 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 - 2652 T + 14404162 T^{2} - 2652 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 1812 T - 9186866 T^{2} - 1812 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 10918 T + 87033783 T^{2} - 10918 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 3316 T + 133272078 T^{2} + 3316 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 19080 T + 322497202 T^{2} + 19080 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 + 12262 T + 310412847 T^{2} + 12262 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 15252 T + 386209090 T^{2} + 15252 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 - 20784 T + 258314650 T^{2} - 20784 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 + 18708 T + 24733646 p T^{2} + 18708 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 35734 T + 1963969491 T^{2} - 35734 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 - 98162 T + 5108967975 T^{2} - 98162 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 + 114120 T + 6722552302 T^{2} + 114120 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 + 109876 T + 6502978230 T^{2} + 109876 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 - 95536 T + 7260163422 T^{2} - 95536 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 - 3732 T + 856206442 T^{2} - 3732 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 93600 T + 12919274098 T^{2} + 93600 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 - 91886 T + 16163058963 T^{2} - 91886 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.059378074473401718522291901389, −8.651019904189100464906892614887, −8.183869995799523271064795447888, −8.171681309455497937424879392788, −7.46283491804060302503512644355, −6.93970589681501820419533169945, −6.62212394475155023752539374626, −6.26696545416944167129545242941, −5.47122229846635955138204961367, −5.34240824882673633972297664976, −4.65346651795094176677449636682, −4.47472395428627716795041254229, −3.47341994619888123025227796589, −3.35480864821603634137842663063, −2.60941988616092604642317340866, −2.31667957692507180839250479226, −1.32949060373189231350106413096, −1.12188294920369126566754634216, 0, 0, 1.12188294920369126566754634216, 1.32949060373189231350106413096, 2.31667957692507180839250479226, 2.60941988616092604642317340866, 3.35480864821603634137842663063, 3.47341994619888123025227796589, 4.47472395428627716795041254229, 4.65346651795094176677449636682, 5.34240824882673633972297664976, 5.47122229846635955138204961367, 6.26696545416944167129545242941, 6.62212394475155023752539374626, 6.93970589681501820419533169945, 7.46283491804060302503512644355, 8.171681309455497937424879392788, 8.183869995799523271064795447888, 8.651019904189100464906892614887, 9.059378074473401718522291901389

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