Properties

Label 4-960e2-1.1-c1e2-0-49
Degree $4$
Conductor $921600$
Sign $1$
Analytic cond. $58.7620$
Root an. cond. $2.76868$
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  + 4·5-s − 9-s + 12·11-s + 8·19-s + 11·25-s − 16·31-s − 12·41-s − 4·45-s + 10·49-s + 48·55-s − 12·59-s + 12·61-s − 8·71-s + 16·79-s + 81-s − 28·89-s + 32·95-s − 12·99-s − 16·101-s + 4·109-s + 86·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯
L(s)  = 1  + 1.78·5-s − 1/3·9-s + 3.61·11-s + 1.83·19-s + 11/5·25-s − 2.87·31-s − 1.87·41-s − 0.596·45-s + 10/7·49-s + 6.47·55-s − 1.56·59-s + 1.53·61-s − 0.949·71-s + 1.80·79-s + 1/9·81-s − 2.96·89-s + 3.28·95-s − 1.20·99-s − 1.59·101-s + 0.383·109-s + 7.81·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-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: $1$
Analytic conductor: \(58.7620\)
Root analytic conductor: \(2.76868\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 921600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.133317576\)
\(L(\frac12)\) \(\approx\) \(4.133317576\)
\(L(\frac{3}{2})\) 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$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 78 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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

−9.891090246945847408836289396824, −9.823698858775263340021678603640, −9.322197807557015415522437040851, −9.012313287191673627759777057095, −8.973088784721920661417098146153, −8.399025752931117581040645304471, −7.53468707191879779860011800857, −7.09124254684315809954009350636, −6.71146342293275352856367529644, −6.54319991819423379909336753817, −5.76595535155158674941118526765, −5.67539042798318409561660598858, −5.19722763649408056950288916249, −4.49614591677651830024462578888, −3.73110619456209469904519566295, −3.62191532841093156722340039772, −2.89485566438997365157697238715, −1.95038766293079945737902604749, −1.52916421218440876802963154580, −1.12940573186945381686729139236, 1.12940573186945381686729139236, 1.52916421218440876802963154580, 1.95038766293079945737902604749, 2.89485566438997365157697238715, 3.62191532841093156722340039772, 3.73110619456209469904519566295, 4.49614591677651830024462578888, 5.19722763649408056950288916249, 5.67539042798318409561660598858, 5.76595535155158674941118526765, 6.54319991819423379909336753817, 6.71146342293275352856367529644, 7.09124254684315809954009350636, 7.53468707191879779860011800857, 8.399025752931117581040645304471, 8.973088784721920661417098146153, 9.012313287191673627759777057095, 9.322197807557015415522437040851, 9.823698858775263340021678603640, 9.891090246945847408836289396824

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