Properties

Label 4-1078e2-1.1-c1e2-0-5
Degree $4$
Conductor $1162084$
Sign $1$
Analytic cond. $74.0954$
Root an. cond. $2.93391$
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 + 2·9-s − 2·11-s + 5·16-s − 4·18-s + 4·22-s + 16·23-s − 10·25-s + 4·29-s − 6·32-s + 6·36-s + 4·37-s − 8·43-s − 6·44-s − 32·46-s + 20·50-s + 28·53-s − 8·58-s + 7·64-s + 8·67-s − 8·72-s − 8·74-s − 32·79-s − 5·81-s + 16·86-s + ⋯
L(s)  = 1  − 1.41·2-s + 3/2·4-s − 1.41·8-s + 2/3·9-s − 0.603·11-s + 5/4·16-s − 0.942·18-s + 0.852·22-s + 3.33·23-s − 2·25-s + 0.742·29-s − 1.06·32-s + 36-s + 0.657·37-s − 1.21·43-s − 0.904·44-s − 4.71·46-s + 2.82·50-s + 3.84·53-s − 1.05·58-s + 7/8·64-s + 0.977·67-s − 0.942·72-s − 0.929·74-s − 3.60·79-s − 5/9·81-s + 1.72·86-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1162084\)    =    \(2^{2} \cdot 7^{4} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(74.0954\)
Root analytic conductor: \(2.93391\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1162084,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.093366089\)
\(L(\frac12)\) \(\approx\) \(1.093366089\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 + T )^{2} \)
7 \( 1 \)
11$C_1$ \( ( 1 + T )^{2} \)
good3$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 74 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 46 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 122 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−10.15102943081488083975851052662, −9.820477154514011232455265934263, −9.085071669556111781842938306281, −8.997466491481587343659289507775, −8.326665340597870716505581679439, −8.323512168724819106043694919347, −7.51717193146548534533156801639, −7.12777322153850334744972367874, −7.11495213396485370016657674131, −6.57607578848459060132495536320, −5.74322054719044137467138824659, −5.66182867008759703605509091147, −4.93923468248198102518591675272, −4.44963573958408879388959489379, −3.78191894378239826821174074022, −3.12631326230837995645522018306, −2.65856383754343291430511836413, −2.05342451150648383206258973081, −1.28019551589000328596134817038, −0.65260617905073516627830173437, 0.65260617905073516627830173437, 1.28019551589000328596134817038, 2.05342451150648383206258973081, 2.65856383754343291430511836413, 3.12631326230837995645522018306, 3.78191894378239826821174074022, 4.44963573958408879388959489379, 4.93923468248198102518591675272, 5.66182867008759703605509091147, 5.74322054719044137467138824659, 6.57607578848459060132495536320, 7.11495213396485370016657674131, 7.12777322153850334744972367874, 7.51717193146548534533156801639, 8.323512168724819106043694919347, 8.326665340597870716505581679439, 8.997466491481587343659289507775, 9.085071669556111781842938306281, 9.820477154514011232455265934263, 10.15102943081488083975851052662

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