Properties

Label 4-576000-1.1-c1e2-0-23
Degree $4$
Conductor $576000$
Sign $1$
Analytic cond. $36.7262$
Root an. cond. $2.46175$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 5-s + 3·9-s + 12·13-s + 2·15-s + 25-s + 4·27-s − 16·31-s − 4·37-s + 24·39-s − 12·41-s + 24·43-s + 3·45-s − 14·49-s + 12·53-s + 12·65-s + 8·67-s + 16·71-s + 2·75-s − 16·79-s + 5·81-s − 24·83-s + 20·89-s − 32·93-s − 8·107-s − 8·111-s + 36·117-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.447·5-s + 9-s + 3.32·13-s + 0.516·15-s + 1/5·25-s + 0.769·27-s − 2.87·31-s − 0.657·37-s + 3.84·39-s − 1.87·41-s + 3.65·43-s + 0.447·45-s − 2·49-s + 1.64·53-s + 1.48·65-s + 0.977·67-s + 1.89·71-s + 0.230·75-s − 1.80·79-s + 5/9·81-s − 2.63·83-s + 2.11·89-s − 3.31·93-s − 0.773·107-s − 0.759·111-s + 3.32·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(576000\)    =    \(2^{9} \cdot 3^{2} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(36.7262\)
Root analytic conductor: \(2.46175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 576000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.077101115\)
\(L(\frac12)\) \(\approx\) \(4.077101115\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( 1 - T \)
good7$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2$ \( ( 1 + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.323057177226078640880314002796, −8.272894657123258864282541087163, −7.66606125131759218514172865215, −6.89270810769313104844169224865, −6.88205984732646038789481985379, −6.10521792180854763107561075146, −5.59585158632682227968586342616, −5.47564398271108061480499478420, −4.38946567062600160988680606034, −3.95303834850677407956821293662, −3.48838247700931708875876876915, −3.25163337009512069140834562852, −2.25002432004047770108316173263, −1.70048286389009539990984644895, −1.09686843848459544952208950048, 1.09686843848459544952208950048, 1.70048286389009539990984644895, 2.25002432004047770108316173263, 3.25163337009512069140834562852, 3.48838247700931708875876876915, 3.95303834850677407956821293662, 4.38946567062600160988680606034, 5.47564398271108061480499478420, 5.59585158632682227968586342616, 6.10521792180854763107561075146, 6.88205984732646038789481985379, 6.89270810769313104844169224865, 7.66606125131759218514172865215, 8.272894657123258864282541087163, 8.323057177226078640880314002796

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