Properties

Label 4-882e2-1.1-c1e2-0-20
Degree $4$
Conductor $777924$
Sign $1$
Analytic cond. $49.6011$
Root an. cond. $2.65382$
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  − 2·2-s + 3·4-s − 4·8-s + 4·11-s + 5·16-s − 8·22-s + 8·23-s − 2·25-s − 4·29-s − 6·32-s + 20·37-s + 4·43-s + 12·44-s − 16·46-s + 4·50-s + 4·53-s + 8·58-s + 7·64-s + 24·67-s + 24·71-s − 40·74-s − 8·79-s − 8·86-s − 16·88-s + 24·92-s − 6·100-s − 8·106-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.20·11-s + 5/4·16-s − 1.70·22-s + 1.66·23-s − 2/5·25-s − 0.742·29-s − 1.06·32-s + 3.28·37-s + 0.609·43-s + 1.80·44-s − 2.35·46-s + 0.565·50-s + 0.549·53-s + 1.05·58-s + 7/8·64-s + 2.93·67-s + 2.84·71-s − 4.64·74-s − 0.900·79-s − 0.862·86-s − 1.70·88-s + 2.50·92-s − 3/5·100-s − 0.777·106-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(777924\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(49.6011\)
Root analytic conductor: \(2.65382\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{882} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 777924,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.264642567\)
\(L(\frac12)\) \(\approx\) \(1.264642567\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
3 \( 1 \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + p T^{2} )^{2} \)
17$C_2^2$ \( 1 + 32 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 12 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 - 16 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 116 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 114 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 128 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 + 96 T^{2} + p^{2} T^{4} \)
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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

−10.09862672275899663901852392365, −9.769944579881218004385660802753, −9.495341183895255823792238557222, −9.197385307516027437701847108737, −8.714418903789153403984824790075, −8.365171298233677812539811164957, −7.79859146866711010455637679558, −7.55772417000874592739120469918, −6.88060638619669399602089114130, −6.78307803372587138832112294560, −6.07270303300013490781867132820, −5.87224703261557223156336786358, −5.09921554447328176462041806446, −4.57508433531885028891043293554, −3.78906696726459253802734177551, −3.52686130493695088434950481777, −2.51699554533182135082823666397, −2.29850546468802787455292500548, −1.20638853268741970222433572774, −0.814577255811514319364394945426, 0.814577255811514319364394945426, 1.20638853268741970222433572774, 2.29850546468802787455292500548, 2.51699554533182135082823666397, 3.52686130493695088434950481777, 3.78906696726459253802734177551, 4.57508433531885028891043293554, 5.09921554447328176462041806446, 5.87224703261557223156336786358, 6.07270303300013490781867132820, 6.78307803372587138832112294560, 6.88060638619669399602089114130, 7.55772417000874592739120469918, 7.79859146866711010455637679558, 8.365171298233677812539811164957, 8.714418903789153403984824790075, 9.197385307516027437701847108737, 9.495341183895255823792238557222, 9.769944579881218004385660802753, 10.09862672275899663901852392365

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