Properties

Degree $4$
Conductor $921600$
Sign $1$
Motivic weight $1$
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·5-s − 9-s + 4·11-s − 16·19-s + 11·25-s + 16·29-s + 4·41-s + 4·45-s + 10·49-s − 16·55-s + 12·59-s − 4·61-s + 8·71-s − 16·79-s + 81-s − 12·89-s + 64·95-s − 4·99-s − 12·109-s − 10·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 64·145-s + 149-s + ⋯
L(s)  = 1  − 1.78·5-s − 1/3·9-s + 1.20·11-s − 3.67·19-s + 11/5·25-s + 2.97·29-s + 0.624·41-s + 0.596·45-s + 10/7·49-s − 2.15·55-s + 1.56·59-s − 0.512·61-s + 0.949·71-s − 1.80·79-s + 1/9·81-s − 1.27·89-s + 6.56·95-s − 0.402·99-s − 1.14·109-s − 0.909·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 5.31·145-s + 0.0819·149-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 921600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(921600\)    =    \(2^{12} \cdot 3^{2} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: induced by $\chi_{960} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.930390\)
\(L(\frac12)\) \(\approx\) \(0.930390\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T^{2} \)
5$C_2$ \( 1 + 4 T + p T^{2} \)
good7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 58 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 130 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−10.26870671570928651544769329884, −10.08363037580864116350997436415, −9.005182915485349800402921961099, −8.938326430419058969539559281165, −8.524708963394712764702786462144, −8.263121450869900726581895515883, −7.88366258694318355537920330272, −7.25810532710205171025243442797, −6.69776241199813181284086333673, −6.51494980513937177174276862019, −6.25788656631927173474389913661, −5.40303945019765498410426269798, −4.73018368441145937045797029115, −4.29048960202738611119589370171, −4.04529255878406932176264212297, −3.75625188599162295958960767058, −2.66281176407388149517619154979, −2.58222617141465845178748427393, −1.38493726723817759877125434325, −0.47867887922774400161696325136, 0.47867887922774400161696325136, 1.38493726723817759877125434325, 2.58222617141465845178748427393, 2.66281176407388149517619154979, 3.75625188599162295958960767058, 4.04529255878406932176264212297, 4.29048960202738611119589370171, 4.73018368441145937045797029115, 5.40303945019765498410426269798, 6.25788656631927173474389913661, 6.51494980513937177174276862019, 6.69776241199813181284086333673, 7.25810532710205171025243442797, 7.88366258694318355537920330272, 8.263121450869900726581895515883, 8.524708963394712764702786462144, 8.938326430419058969539559281165, 9.005182915485349800402921961099, 10.08363037580864116350997436415, 10.26870671570928651544769329884

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