Properties

Label 4-1200e2-1.1-c3e2-0-16
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $5012.96$
Root an. cond. $8.41440$
Motivic weight $3$
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  − 9·9-s + 96·11-s + 280·19-s − 420·29-s − 544·31-s − 396·41-s + 670·49-s + 480·59-s + 604·61-s + 1.53e3·71-s − 1.28e3·79-s + 81·81-s − 420·89-s − 864·99-s + 3.44e3·101-s + 1.22e3·109-s + 4.25e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 4.39e3·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 2.63·11-s + 3.38·19-s − 2.68·29-s − 3.15·31-s − 1.50·41-s + 1.95·49-s + 1.05·59-s + 1.26·61-s + 2.56·71-s − 1.82·79-s + 1/9·81-s − 0.500·89-s − 0.877·99-s + 3.39·101-s + 1.07·109-s + 3.19·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 1.99·169-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(5012.96\)
Root analytic conductor: \(8.41440\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(4.258995545\)
\(L(\frac12)\) \(\approx\) \(4.258995545\)
\(L(\frac{5}{2})\) 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$C_2$ \( 1 + p^{2} T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - 670 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 48 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 - 4390 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 + 3170 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 - 140 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 19150 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 + 210 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 272 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 + 10250 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 198 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 87190 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 160990 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 291670 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 240 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 - 302 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 246310 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 768 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 549550 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 640 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1022470 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 210 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 + 527810 T^{2} + p^{6} T^{4} \)
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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

−9.359799366833261840636254597388, −9.312330133089882012630771800149, −8.871559736343740868445159615325, −8.605876664799966058516507915818, −7.72943548797226142481689761762, −7.44456820201042208779106791273, −7.05710802281960569950249482998, −6.94910197208657226420546753877, −6.19560886629945639039652644543, −5.63641496991596172262758155199, −5.34628656762851323706343060284, −5.23391451551795439624459672109, −4.05721486601111057134018429552, −3.92338759499866711659654893239, −3.32786197715328077176298711000, −3.31210587949333384787610363608, −1.94103194222554993707145549004, −1.83957999922518805068485831117, −1.05292629844596158041362267066, −0.56398853839584598481768331207, 0.56398853839584598481768331207, 1.05292629844596158041362267066, 1.83957999922518805068485831117, 1.94103194222554993707145549004, 3.31210587949333384787610363608, 3.32786197715328077176298711000, 3.92338759499866711659654893239, 4.05721486601111057134018429552, 5.23391451551795439624459672109, 5.34628656762851323706343060284, 5.63641496991596172262758155199, 6.19560886629945639039652644543, 6.94910197208657226420546753877, 7.05710802281960569950249482998, 7.44456820201042208779106791273, 7.72943548797226142481689761762, 8.605876664799966058516507915818, 8.871559736343740868445159615325, 9.312330133089882012630771800149, 9.359799366833261840636254597388

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