Properties

Label 4-1200e2-1.1-c3e2-0-11
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 9·9-s − 84·11-s − 230·19-s + 420·29-s + 386·31-s + 24·41-s + 685·49-s + 1.38e3·59-s − 1.46e3·61-s + 456·71-s − 320·79-s + 81·81-s + 480·89-s + 756·99-s + 1.82e3·101-s + 3.47e3·109-s + 2.63e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 95·169-s + ⋯
L(s)  = 1  − 1/3·9-s − 2.30·11-s − 2.77·19-s + 2.68·29-s + 2.23·31-s + 0.0914·41-s + 1.99·49-s + 3.04·59-s − 3.07·61-s + 0.762·71-s − 0.455·79-s + 1/9·81-s + 0.571·89-s + 0.767·99-s + 1.79·101-s + 3.04·109-s + 1.97·121-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 − 0.0432·169-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

\(L(2)\) \(\approx\) \(2.079500385\)
\(L(\frac12)\) \(\approx\) \(2.079500385\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 685 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 + 42 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 95 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 6910 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 115 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 + 1910 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 210 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 - 193 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 19510 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 12 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 89845 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 36250 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 260890 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 690 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 733 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 512125 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 228 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 101810 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 160 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 930130 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 - 240 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1564225 T^{2} + p^{6} T^{4} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.761836937519103171611012461788, −8.945668163689817061408387429813, −8.541474982401347227654014230095, −8.463137815332931926072856704930, −8.078744092627659192562006052269, −7.62798733774925553933358993244, −7.12148276795583232838510876792, −6.57555185584490663999418252963, −6.18221641425964079490542097212, −5.97952787468616285686080369476, −5.22462872984904700318983263336, −4.86019748160008048949820800743, −4.45896263066938463554294840083, −4.12690351616618969844920618144, −3.21226214650368216767803415266, −2.64437790840564291877786052782, −2.51198740043673519225312159829, −1.95611203160081603962564116070, −0.77327502140195424191626100072, −0.46448602788039276588287141850, 0.46448602788039276588287141850, 0.77327502140195424191626100072, 1.95611203160081603962564116070, 2.51198740043673519225312159829, 2.64437790840564291877786052782, 3.21226214650368216767803415266, 4.12690351616618969844920618144, 4.45896263066938463554294840083, 4.86019748160008048949820800743, 5.22462872984904700318983263336, 5.97952787468616285686080369476, 6.18221641425964079490542097212, 6.57555185584490663999418252963, 7.12148276795583232838510876792, 7.62798733774925553933358993244, 8.078744092627659192562006052269, 8.463137815332931926072856704930, 8.541474982401347227654014230095, 8.945668163689817061408387429813, 9.761836937519103171611012461788

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