Properties

Label 4-1200e2-1.1-c3e2-0-3
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 9·9-s + 48·11-s − 248·19-s + 156·29-s − 400·31-s + 660·41-s + 286·49-s + 48·59-s − 644·61-s + 576·71-s − 1.04e3·79-s + 81·81-s − 2.05e3·89-s − 432·99-s − 3.46e3·101-s + 2.94e3·109-s − 934·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 1.08e3·169-s + ⋯
L(s)  = 1  − 1/3·9-s + 1.31·11-s − 2.99·19-s + 0.998·29-s − 2.31·31-s + 2.51·41-s + 0.833·49-s + 0.105·59-s − 1.35·61-s + 0.962·71-s − 1.48·79-s + 1/9·81-s − 2.44·89-s − 0.438·99-s − 3.41·101-s + 2.59·109-s − 0.701·121-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 − 0.492·169-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: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :3/2, 3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7957252257\)
\(L(\frac12)\) \(\approx\) \(0.7957252257\)
\(L(\frac{5}{2})\) 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$C_2^2$ \( 1 - 286 T^{2} + p^{6} T^{4} \)
11$C_2$ \( ( 1 - 24 T + p^{3} T^{2} )^{2} \)
13$C_2^2$ \( 1 + 1082 T^{2} + p^{6} T^{4} \)
17$C_2^2$ \( 1 - 6910 T^{2} + p^{6} T^{4} \)
19$C_2$ \( ( 1 + 124 T + p^{3} T^{2} )^{2} \)
23$C_2^2$ \( 1 - 9934 T^{2} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 78 T + p^{3} T^{2} )^{2} \)
31$C_2$ \( ( 1 + 200 T + p^{3} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 96406 T^{2} + p^{6} T^{4} \)
41$C_2$ \( ( 1 - 330 T + p^{3} T^{2} )^{2} \)
43$C_2^2$ \( 1 - 150550 T^{2} + p^{6} T^{4} \)
47$C_2^2$ \( 1 - 207070 T^{2} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 95254 T^{2} + p^{6} T^{4} \)
59$C_2$ \( ( 1 - 24 T + p^{3} T^{2} )^{2} \)
61$C_2$ \( ( 1 + 322 T + p^{3} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 563110 T^{2} + p^{6} T^{4} \)
71$C_2$ \( ( 1 - 288 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 - 593134 T^{2} + p^{6} T^{4} \)
79$C_2$ \( ( 1 + 520 T + p^{3} T^{2} )^{2} \)
83$C_2^2$ \( 1 - 1119238 T^{2} + p^{6} T^{4} \)
89$C_2$ \( ( 1 + 1026 T + p^{3} T^{2} )^{2} \)
97$C_2^2$ \( 1 - 1743550 T^{2} + p^{6} T^{4} \)
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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

−9.477869427899281601708240070049, −9.190427115456054174595655763355, −8.651835109093920234002452689501, −8.527548874405381277891954754166, −8.103260010753045588067054210085, −7.30557146623370923150234511029, −7.23430904923972942689520385288, −6.47395921539316655316216971336, −6.41331688706029227602565090476, −5.70177952596485951564210421818, −5.65462286231385025014553171046, −4.61159207576095690926885741894, −4.43074428845688217207928755300, −3.89548967951828831462284507502, −3.65130791461614091271262633170, −2.63722341833104839325747684010, −2.43084291535131809630774653369, −1.67509066677407199409440278082, −1.16956726109639021383296214670, −0.21732117931519244774116973146, 0.21732117931519244774116973146, 1.16956726109639021383296214670, 1.67509066677407199409440278082, 2.43084291535131809630774653369, 2.63722341833104839325747684010, 3.65130791461614091271262633170, 3.89548967951828831462284507502, 4.43074428845688217207928755300, 4.61159207576095690926885741894, 5.65462286231385025014553171046, 5.70177952596485951564210421818, 6.41331688706029227602565090476, 6.47395921539316655316216971336, 7.23430904923972942689520385288, 7.30557146623370923150234511029, 8.103260010753045588067054210085, 8.527548874405381277891954754166, 8.651835109093920234002452689501, 9.190427115456054174595655763355, 9.477869427899281601708240070049

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