Properties

Label 4-1568e2-1.1-c1e2-0-13
Degree $4$
Conductor $2458624$
Sign $1$
Analytic cond. $156.763$
Root an. cond. $3.53843$
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  − 4·9-s + 4·11-s + 16·23-s − 10·25-s + 12·29-s − 4·37-s + 12·43-s + 12·53-s + 24·67-s − 8·71-s + 24·79-s + 7·81-s − 16·99-s + 8·107-s + 36·109-s − 24·113-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 18·169-s + ⋯
L(s)  = 1  − 4/3·9-s + 1.20·11-s + 3.33·23-s − 2·25-s + 2.22·29-s − 0.657·37-s + 1.82·43-s + 1.64·53-s + 2.93·67-s − 0.949·71-s + 2.70·79-s + 7/9·81-s − 1.60·99-s + 0.773·107-s + 3.44·109-s − 2.25·113-s − 0.909·121-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 − 1.38·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2458624 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2458624\)    =    \(2^{10} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(156.763\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2458624,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.575123429\)
\(L(\frac12)\) \(\approx\) \(2.575123429\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 20 T^{2} + p^{2} T^{4} \)
23$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2^2$ \( 1 + 64 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 44 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 144 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 68 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 160 T^{2} + p^{2} T^{4} \)
97$C_2^2$ \( 1 - 144 T^{2} + p^{2} T^{4} \)
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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

−9.316136142031300408698804928428, −9.282302887481105088766968489939, −8.895128552716611534346848357227, −8.473114568179355300384736496876, −8.194046697109229900159273724512, −7.65175389698849227106722506071, −7.03152444091693357187156538595, −6.95543157337746714496926862097, −6.26071282459919080446857332955, −6.15669626114683451886951610155, −5.34178652615149445904380531671, −5.29868659239306167819656585582, −4.63425513534223891106207765918, −4.18197883597665849303106895228, −3.44271357763943207390107548034, −3.35551205628358491016998314902, −2.48258315435609128359806847864, −2.29772124931793293097997979003, −1.12552427816405528035864022651, −0.75407052042068419041052498289, 0.75407052042068419041052498289, 1.12552427816405528035864022651, 2.29772124931793293097997979003, 2.48258315435609128359806847864, 3.35551205628358491016998314902, 3.44271357763943207390107548034, 4.18197883597665849303106895228, 4.63425513534223891106207765918, 5.29868659239306167819656585582, 5.34178652615149445904380531671, 6.15669626114683451886951610155, 6.26071282459919080446857332955, 6.95543157337746714496926862097, 7.03152444091693357187156538595, 7.65175389698849227106722506071, 8.194046697109229900159273724512, 8.473114568179355300384736496876, 8.895128552716611534346848357227, 9.282302887481105088766968489939, 9.316136142031300408698804928428

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