Properties

Label 4-80e4-1.1-c1e2-0-31
Degree $4$
Conductor $40960000$
Sign $1$
Analytic cond. $2611.64$
Root an. cond. $7.14872$
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  + 2·3-s − 2·7-s − 4·11-s − 8·19-s − 4·21-s − 2·23-s − 2·27-s − 4·31-s − 8·33-s + 4·37-s + 4·41-s + 14·43-s − 10·47-s − 8·49-s − 16·53-s − 16·57-s − 8·59-s + 4·61-s + 18·67-s − 4·69-s − 4·71-s + 8·73-s + 8·77-s − 16·79-s − 81-s + 6·83-s − 4·89-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s − 1.20·11-s − 1.83·19-s − 0.872·21-s − 0.417·23-s − 0.384·27-s − 0.718·31-s − 1.39·33-s + 0.657·37-s + 0.624·41-s + 2.13·43-s − 1.45·47-s − 8/7·49-s − 2.19·53-s − 2.11·57-s − 1.04·59-s + 0.512·61-s + 2.19·67-s − 0.481·69-s − 0.474·71-s + 0.936·73-s + 0.911·77-s − 1.80·79-s − 1/9·81-s + 0.658·83-s − 0.423·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40960000\)    =    \(2^{16} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2611.64\)
Root analytic conductor: \(7.14872\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 40960000,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 4 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 12 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 2 T + 20 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 74 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 14 T + 132 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 116 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 16 T + 158 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 8 T + 122 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 18 T + 212 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 6 T + 172 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 4 T + 134 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−7.79161991245896868881774449253, −7.77237609638573942369887945976, −7.31339575413653263671036873607, −6.65559154866201543404930637388, −6.40822026763558956695515942464, −6.30328675373665937973372476588, −5.70335793892666390632137277159, −5.42747754831506426406829706361, −4.86955597835043821188753150976, −4.60864467401468554082226665843, −3.99317622618812923927802151972, −3.88953297959578439774329447287, −3.12529077152369774723408802073, −3.07604351301349161789284465520, −2.55605404189965930725500557986, −2.22801574052017612220871547054, −1.85217003932706373666378857964, −1.08054165479826905556508312822, 0, 0, 1.08054165479826905556508312822, 1.85217003932706373666378857964, 2.22801574052017612220871547054, 2.55605404189965930725500557986, 3.07604351301349161789284465520, 3.12529077152369774723408802073, 3.88953297959578439774329447287, 3.99317622618812923927802151972, 4.60864467401468554082226665843, 4.86955597835043821188753150976, 5.42747754831506426406829706361, 5.70335793892666390632137277159, 6.30328675373665937973372476588, 6.40822026763558956695515942464, 6.65559154866201543404930637388, 7.31339575413653263671036873607, 7.77237609638573942369887945976, 7.79161991245896868881774449253

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