Properties

Degree $4$
Conductor $1382976$
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  − 2·3-s + 4·5-s + 3·9-s − 4·11-s − 8·15-s + 12·17-s + 8·19-s − 4·23-s + 4·25-s − 4·27-s + 8·33-s + 8·37-s + 12·41-s + 12·45-s + 8·47-s − 24·51-s − 4·53-s − 16·55-s − 16·57-s + 8·61-s + 8·69-s − 4·71-s + 24·73-s − 8·75-s + 16·79-s + 5·81-s − 8·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 1.78·5-s + 9-s − 1.20·11-s − 2.06·15-s + 2.91·17-s + 1.83·19-s − 0.834·23-s + 4/5·25-s − 0.769·27-s + 1.39·33-s + 1.31·37-s + 1.87·41-s + 1.78·45-s + 1.16·47-s − 3.36·51-s − 0.549·53-s − 2.15·55-s − 2.11·57-s + 1.02·61-s + 0.963·69-s − 0.474·71-s + 2.80·73-s − 0.923·75-s + 1.80·79-s + 5/9·81-s − 0.878·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1382976\)    =    \(2^{6} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{1176} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1382976,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.568199801\)
\(L(\frac12)\) \(\approx\) \(2.568199801\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
good5$D_{4}$ \( 1 - 4 T + 12 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 8 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 12 T + 4 p T^{2} - 12 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
23$C_4$ \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
31$C_2^2$ \( 1 + 54 T^{2} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 12 T + 100 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
43$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 8 T + 88 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 24 T + 272 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 16 T + 190 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 20 T + 260 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 8 T + 192 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−9.957739283219519194481216297863, −9.874999623649005635059614898504, −9.355554026761523516885561775349, −9.070436876700283002549273606440, −8.022680413609285129565670462565, −7.87350155908191225899270462910, −7.52823551613894385119104245611, −7.19636794737525610329331931747, −6.27285204175584493663561643222, −6.03441000081613018251875105271, −5.79067913139686146174150171718, −5.44350299069127547691562540748, −5.06627991215715855586940941927, −4.73148377689358237311182151985, −3.61151833613487952017859474691, −3.50767919790606919660421938774, −2.42875513623203969810029701258, −2.29629771145379592239840068789, −1.14577477740938300051393507152, −0.932550693858387367066718846611, 0.932550693858387367066718846611, 1.14577477740938300051393507152, 2.29629771145379592239840068789, 2.42875513623203969810029701258, 3.50767919790606919660421938774, 3.61151833613487952017859474691, 4.73148377689358237311182151985, 5.06627991215715855586940941927, 5.44350299069127547691562540748, 5.79067913139686146174150171718, 6.03441000081613018251875105271, 6.27285204175584493663561643222, 7.19636794737525610329331931747, 7.52823551613894385119104245611, 7.87350155908191225899270462910, 8.022680413609285129565670462565, 9.070436876700283002549273606440, 9.355554026761523516885561775349, 9.874999623649005635059614898504, 9.957739283219519194481216297863

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