Properties

Label 4-1920e2-1.1-c1e2-0-10
Degree $4$
Conductor $3686400$
Sign $1$
Analytic cond. $235.048$
Root an. cond. $3.91551$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·5-s − 9-s − 12·19-s + 11·25-s + 16·29-s − 16·31-s − 12·41-s − 4·45-s − 2·49-s + 24·59-s + 28·61-s − 16·71-s + 24·79-s + 81-s − 12·89-s − 48·95-s + 4·109-s − 22·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s − 64·155-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s − 2.75·19-s + 11/5·25-s + 2.97·29-s − 2.87·31-s − 1.87·41-s − 0.596·45-s − 2/7·49-s + 3.12·59-s + 3.58·61-s − 1.89·71-s + 2.70·79-s + 1/9·81-s − 1.27·89-s − 4.92·95-s + 0.383·109-s − 2·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s − 5.14·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3686400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(3686400\)    =    \(2^{14} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(235.048\)
Root analytic conductor: \(3.91551\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 3686400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.741740558\)
\(L(\frac12)\) \(\approx\) \(2.741740558\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 90 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−9.406171627091146240017670032624, −8.852788276641653995601695500925, −8.677837639968309451837005472450, −8.392865466590526959443821282965, −8.121732002480809353675107529992, −7.21516605585194194959190067794, −6.79989865345280484695001224396, −6.62294726370191494405706318762, −6.37911386646073402901520764011, −5.63860816953378232698023246273, −5.57182273236609865220494082500, −4.95065225135551634082598349079, −4.73602281260995820595505859434, −3.83291737925194375357772197923, −3.76119844539937844423297141054, −2.79139402660676206332875153889, −2.39645388668815366288858194558, −2.05373433774555767093572023145, −1.52897513463252493945054421534, −0.58184425971407767233654672679, 0.58184425971407767233654672679, 1.52897513463252493945054421534, 2.05373433774555767093572023145, 2.39645388668815366288858194558, 2.79139402660676206332875153889, 3.76119844539937844423297141054, 3.83291737925194375357772197923, 4.73602281260995820595505859434, 4.95065225135551634082598349079, 5.57182273236609865220494082500, 5.63860816953378232698023246273, 6.37911386646073402901520764011, 6.62294726370191494405706318762, 6.79989865345280484695001224396, 7.21516605585194194959190067794, 8.121732002480809353675107529992, 8.392865466590526959443821282965, 8.677837639968309451837005472450, 8.852788276641653995601695500925, 9.406171627091146240017670032624

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