Properties

Label 4-30e4-1.1-c1e2-0-5
Degree $4$
Conductor $810000$
Sign $1$
Analytic cond. $51.6463$
Root an. cond. $2.68077$
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  + 8·19-s + 12·29-s − 8·31-s − 12·41-s + 10·49-s + 24·59-s + 4·61-s + 24·71-s − 16·79-s − 12·89-s − 12·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 1.83·19-s + 2.22·29-s − 1.43·31-s − 1.87·41-s + 10/7·49-s + 3.12·59-s + 0.512·61-s + 2.84·71-s − 1.80·79-s − 1.27·89-s − 1.19·101-s − 0.383·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(810000\)    =    \(2^{4} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(51.6463\)
Root analytic conductor: \(2.68077\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 810000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.127225318\)
\(L(\frac12)\) \(\approx\) \(2.127225318\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−10.15439932560684357266135977229, −10.01570811954058452992744038126, −9.489725898311464400300868502613, −9.156164047109138511474296616246, −8.482322343258797703440780631439, −8.326502679820198834208779532300, −7.907163243023524331780009047822, −7.16773504798249208200477013674, −6.84957622951006660848628738644, −6.78989105422016221469085682128, −5.81742469470801246302065654171, −5.47809821580915633613967569270, −5.16233867670098546519770859243, −4.62082594621544345676391001556, −3.83796355583813629637042847422, −3.60318739957152218051263604951, −2.84618538873820592025711437490, −2.39193764197407511426374929168, −1.47187082659401271260004168201, −0.75253694333488070631697540925, 0.75253694333488070631697540925, 1.47187082659401271260004168201, 2.39193764197407511426374929168, 2.84618538873820592025711437490, 3.60318739957152218051263604951, 3.83796355583813629637042847422, 4.62082594621544345676391001556, 5.16233867670098546519770859243, 5.47809821580915633613967569270, 5.81742469470801246302065654171, 6.78989105422016221469085682128, 6.84957622951006660848628738644, 7.16773504798249208200477013674, 7.907163243023524331780009047822, 8.326502679820198834208779532300, 8.482322343258797703440780631439, 9.156164047109138511474296616246, 9.489725898311464400300868502613, 10.01570811954058452992744038126, 10.15439932560684357266135977229

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