Properties

Label 4-120e2-1.1-c1e2-0-0
Degree $4$
Conductor $14400$
Sign $1$
Analytic cond. $0.918156$
Root an. cond. $0.978879$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − 3-s + 2·4-s − 4·5-s + 2·6-s − 2·9-s + 8·10-s − 2·12-s + 4·15-s − 4·16-s + 4·18-s − 5·19-s − 8·20-s + 8·23-s + 11·25-s + 5·27-s − 8·30-s + 8·32-s − 4·36-s + 10·38-s + 8·43-s + 8·45-s − 16·46-s − 4·47-s + 4·48-s + 5·49-s − 22·50-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577·3-s + 4-s − 1.78·5-s + 0.816·6-s − 2/3·9-s + 2.52·10-s − 0.577·12-s + 1.03·15-s − 16-s + 0.942·18-s − 1.14·19-s − 1.78·20-s + 1.66·23-s + 11/5·25-s + 0.962·27-s − 1.46·30-s + 1.41·32-s − 2/3·36-s + 1.62·38-s + 1.21·43-s + 1.19·45-s − 2.35·46-s − 0.583·47-s + 0.577·48-s + 5/7·49-s − 3.11·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(14400\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(0.918156\)
Root analytic conductor: \(0.978879\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 14400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.2399418405\)
\(L(\frac12)\) \(\approx\) \(0.2399418405\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 + T + p T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 5 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 3 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 15 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
41$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \)
79$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
89$C_2^2$ \( 1 + 73 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
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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

−11.10453689483972381529094161403, −10.71483285980562108266019034372, −10.27987266638457484031606556775, −9.295267517386777252505838054624, −8.857571587037774996858240106627, −8.440310595556864067370478411292, −7.955712769100263634022911097959, −7.29016723371577441364561855942, −6.90476787494751312784236348203, −6.17214632757634110654137577125, −5.10899066757796848109208201325, −4.50596100523206304792403612920, −3.65981850259497476626944606770, −2.59159221212238976975669095197, −0.70139200864348904989459807621, 0.70139200864348904989459807621, 2.59159221212238976975669095197, 3.65981850259497476626944606770, 4.50596100523206304792403612920, 5.10899066757796848109208201325, 6.17214632757634110654137577125, 6.90476787494751312784236348203, 7.29016723371577441364561855942, 7.955712769100263634022911097959, 8.440310595556864067370478411292, 8.857571587037774996858240106627, 9.295267517386777252505838054624, 10.27987266638457484031606556775, 10.71483285980562108266019034372, 11.10453689483972381529094161403

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