Properties

Label 4-9280e2-1.1-c1e2-0-5
Degree $4$
Conductor $86118400$
Sign $1$
Analytic cond. $5490.98$
Root an. cond. $8.60820$
Motivic weight $1$
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  − 3-s + 2·5-s − 7-s − 4·9-s − 2·11-s − 13-s − 2·15-s − 5·17-s + 10·19-s + 21-s − 5·23-s + 3·25-s + 6·27-s − 2·29-s + 31-s + 2·33-s − 2·35-s − 14·37-s + 39-s + 6·41-s − 13·43-s − 8·45-s + 16·47-s − 2·49-s + 5·51-s − 3·53-s − 4·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s − 0.377·7-s − 4/3·9-s − 0.603·11-s − 0.277·13-s − 0.516·15-s − 1.21·17-s + 2.29·19-s + 0.218·21-s − 1.04·23-s + 3/5·25-s + 1.15·27-s − 0.371·29-s + 0.179·31-s + 0.348·33-s − 0.338·35-s − 2.30·37-s + 0.160·39-s + 0.937·41-s − 1.98·43-s − 1.19·45-s + 2.33·47-s − 2/7·49-s + 0.700·51-s − 0.412·53-s − 0.539·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86118400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(86118400\)    =    \(2^{12} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(5490.98\)
Root analytic conductor: \(8.60820\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 86118400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
29$C_1$ \( ( 1 + T )^{2} \)
good3$D_{4}$ \( 1 + T + 5 T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + T - 5 T^{2} + p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 5 T + 39 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 5 T + 41 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - T + 61 T^{2} - p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 14 T + 118 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 13 T + 127 T^{2} + 13 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 3 T + 47 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 21 T + 227 T^{2} + 21 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 8 T + 78 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 15 T + 171 T^{2} - 15 p T^{3} + p^{2} T^{4} \)
79$C_4$ \( 1 - 17 T + 199 T^{2} - 17 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 18 T + 202 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 10 T + 158 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 7 T + 55 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
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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

−7.68461120697359545924080599763, −7.18537745321075366240194484741, −6.59071439582338268523087430247, −6.56348919478569228712807666112, −6.05521452428192297044597108363, −5.97185881820493212500492908518, −5.30049705687412399126136835053, −5.29698882399905124388351815679, −4.91820157356936199468046802151, −4.72540277085653130825676729484, −3.77875978655603992300157060482, −3.72674197583322772656483679823, −3.01921857406596505211415376088, −2.98284160532106776929553404740, −2.32737320140875471507430834513, −2.09526798235140453211705159528, −1.49015638559886330831349623774, −0.921539005527916141521636309971, 0, 0, 0.921539005527916141521636309971, 1.49015638559886330831349623774, 2.09526798235140453211705159528, 2.32737320140875471507430834513, 2.98284160532106776929553404740, 3.01921857406596505211415376088, 3.72674197583322772656483679823, 3.77875978655603992300157060482, 4.72540277085653130825676729484, 4.91820157356936199468046802151, 5.29698882399905124388351815679, 5.30049705687412399126136835053, 5.97185881820493212500492908518, 6.05521452428192297044597108363, 6.56348919478569228712807666112, 6.59071439582338268523087430247, 7.18537745321075366240194484741, 7.68461120697359545924080599763

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