Properties

Label 4-4840e2-1.1-c1e2-0-2
Degree $4$
Conductor $23425600$
Sign $1$
Analytic cond. $1493.63$
Root an. cond. $6.21671$
Motivic weight $1$
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  + 2·5-s + 4·7-s − 3·9-s + 4·19-s + 8·23-s + 3·25-s + 4·29-s + 4·31-s + 8·35-s − 8·37-s + 10·41-s − 6·45-s − 4·47-s + 49-s − 8·53-s + 4·59-s + 10·61-s − 12·63-s + 8·67-s + 12·71-s + 16·73-s + 8·79-s − 8·83-s + 18·89-s + 8·95-s − 8·97-s + 2·101-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s − 9-s + 0.917·19-s + 1.66·23-s + 3/5·25-s + 0.742·29-s + 0.718·31-s + 1.35·35-s − 1.31·37-s + 1.56·41-s − 0.894·45-s − 0.583·47-s + 1/7·49-s − 1.09·53-s + 0.520·59-s + 1.28·61-s − 1.51·63-s + 0.977·67-s + 1.42·71-s + 1.87·73-s + 0.900·79-s − 0.878·83-s + 1.90·89-s + 0.820·95-s − 0.812·97-s + 0.199·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23425600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(23425600\)    =    \(2^{6} \cdot 5^{2} \cdot 11^{4}\)
Sign: $1$
Analytic conductor: \(1493.63\)
Root analytic conductor: \(6.21671\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 23425600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.237638134\)
\(L(\frac12)\) \(\approx\) \(5.237638134\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
11 \( 1 \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
23$D_{4}$ \( 1 - 8 T + 50 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 + 83 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 4 T + 110 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 10 T + 135 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 147 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 162 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 8 T + 170 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 211 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
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

−8.334478838327299477908466859843, −8.272488456576756255872574786662, −7.79079735469737812955277319145, −7.40282387736873602965041448527, −7.01509972314152779526802032839, −6.65240158935321571374247941663, −6.10208285811065490028923709807, −6.07473690634459949053463561798, −5.33647778442636584547480486025, −5.06810951990086424050012454881, −4.90720525055885104541809783706, −4.75311790584034692875850759186, −3.74524939751106978373135225266, −3.64410026370568243906604098261, −2.93133344243853521266691580614, −2.64212749338499307005255567665, −2.15697579880512839440493919200, −1.69660833685283916738577004391, −1.04695039319725611217707371457, −0.72410449708954778097275374831, 0.72410449708954778097275374831, 1.04695039319725611217707371457, 1.69660833685283916738577004391, 2.15697579880512839440493919200, 2.64212749338499307005255567665, 2.93133344243853521266691580614, 3.64410026370568243906604098261, 3.74524939751106978373135225266, 4.75311790584034692875850759186, 4.90720525055885104541809783706, 5.06810951990086424050012454881, 5.33647778442636584547480486025, 6.07473690634459949053463561798, 6.10208285811065490028923709807, 6.65240158935321571374247941663, 7.01509972314152779526802032839, 7.40282387736873602965041448527, 7.79079735469737812955277319145, 8.272488456576756255872574786662, 8.334478838327299477908466859843

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