Properties

Degree $4$
Conductor $2286144$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·5-s + 4·7-s − 4·11-s − 2·13-s + 4·19-s + 6·23-s + 5·25-s − 5·31-s + 8·35-s − 37-s + 8·41-s − 2·43-s + 4·47-s + 9·49-s − 6·53-s − 8·55-s + 3·61-s − 4·65-s − 11·67-s + 28·71-s + 14·73-s − 16·77-s − 13·79-s − 28·83-s + 6·89-s − 8·91-s + 8·95-s + ⋯
L(s)  = 1  + 0.894·5-s + 1.51·7-s − 1.20·11-s − 0.554·13-s + 0.917·19-s + 1.25·23-s + 25-s − 0.898·31-s + 1.35·35-s − 0.164·37-s + 1.24·41-s − 0.304·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s − 1.07·55-s + 0.384·61-s − 0.496·65-s − 1.34·67-s + 3.32·71-s + 1.63·73-s − 1.82·77-s − 1.46·79-s − 3.07·83-s + 0.635·89-s − 0.838·91-s + 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2286144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2286144\)    =    \(2^{6} \cdot 3^{6} \cdot 7^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1512} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2286144,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.325158964\)
\(L(\frac12)\) \(\approx\) \(3.325158964\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7$C_2$ \( 1 - 4 T + p T^{2} \)
good5$C_2^2$ \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 5 T - 6 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 4 T - 31 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
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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

−9.823348706204214982254806881826, −9.142562256287727828682841364170, −8.892714363348812050889419324501, −8.602451448696083889421067042262, −7.947966547760633412314276520459, −7.60788226061128595362499722709, −7.48310748097923436102757240735, −6.90126094104091601505541024691, −6.47515135539313978584301072896, −5.77267080365393015011603722200, −5.41376898939079185091165290225, −5.24800358713264851021598547840, −4.69136893603325239482987738266, −4.51214234160735207134550349650, −3.59649481363211597308414624373, −3.07720860368886879697045209395, −2.48534898001024316624520542329, −2.11591438405784893882059971325, −1.44461821182599522797268036039, −0.75772461077159913566858037421, 0.75772461077159913566858037421, 1.44461821182599522797268036039, 2.11591438405784893882059971325, 2.48534898001024316624520542329, 3.07720860368886879697045209395, 3.59649481363211597308414624373, 4.51214234160735207134550349650, 4.69136893603325239482987738266, 5.24800358713264851021598547840, 5.41376898939079185091165290225, 5.77267080365393015011603722200, 6.47515135539313978584301072896, 6.90126094104091601505541024691, 7.48310748097923436102757240735, 7.60788226061128595362499722709, 7.947966547760633412314276520459, 8.602451448696083889421067042262, 8.892714363348812050889419324501, 9.142562256287727828682841364170, 9.823348706204214982254806881826

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