Properties

Label 4-1200e2-1.1-c2e2-0-7
Degree $4$
Conductor $1440000$
Sign $1$
Analytic cond. $1069.13$
Root an. cond. $5.71818$
Motivic weight $2$
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 + 22·19-s − 118·31-s − 71·49-s − 242·61-s − 284·79-s + 81·81-s − 142·109-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 191·169-s − 198·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 9-s + 1.15·19-s − 3.80·31-s − 1.44·49-s − 3.96·61-s − 3.59·79-s + 81-s − 1.30·109-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 1.13·169-s − 1.15·171-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + 0.00518·193-s + 0.00507·197-s + 0.00502·199-s + ⋯

Functional equation

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

Invariants

Degree: \(4\)
Conductor: \(1440000\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(1069.13\)
Root analytic conductor: \(5.71818\)
Motivic weight: \(2\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{1200} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1440000,\ (\ :1, 1),\ 1)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2931500203\)
\(L(\frac12)\) \(\approx\) \(0.2931500203\)
\(L(2)\) 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$C_2^2$ \( 1 + 71 T^{2} + p^{4} T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
13$C_2^2$ \( 1 + 191 T^{2} + p^{4} T^{4} \)
17$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
19$C_2$ \( ( 1 - 11 T + p^{2} T^{2} )^{2} \)
23$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
31$C_2$ \( ( 1 + 59 T + p^{2} T^{2} )^{2} \)
37$C_2^2$ \( 1 - 2062 T^{2} + p^{4} T^{4} \)
41$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
43$C_2^2$ \( 1 + 3191 T^{2} + p^{4} T^{4} \)
47$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
53$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
61$C_2$ \( ( 1 + 121 T + p^{2} T^{2} )^{2} \)
67$C_2^2$ \( 1 - 8809 T^{2} + p^{4} T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
73$C_2^2$ \( 1 - 8542 T^{2} + p^{4} T^{4} \)
79$C_2$ \( ( 1 + 142 T + p^{2} T^{2} )^{2} \)
83$C_2$ \( ( 1 + p^{2} T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - p T )^{2}( 1 + p T )^{2} \)
97$C_2^2$ \( 1 + 9071 T^{2} + p^{4} T^{4} \)
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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

−9.596125079566744108661686525046, −9.389777395671716201513469034411, −8.881024066759561667128452120861, −8.728600786314345023625280564125, −8.114061847201436409623374491730, −7.54488443915516510283392574198, −7.44603916369347235577420959500, −7.00816169936187185627896678697, −6.32804281240571736527096397442, −5.83606009752520577561577590139, −5.66828971538788621540513019141, −5.13650467608733987479659882388, −4.69938630315114602413932672281, −4.07092034965117253693955869920, −3.41512080014130764067236388147, −3.19981813948027966126231581276, −2.62016294013058508604226690154, −1.74377695506693417430781020040, −1.42332371151406899348358766287, −0.15677652851306412910321358092, 0.15677652851306412910321358092, 1.42332371151406899348358766287, 1.74377695506693417430781020040, 2.62016294013058508604226690154, 3.19981813948027966126231581276, 3.41512080014130764067236388147, 4.07092034965117253693955869920, 4.69938630315114602413932672281, 5.13650467608733987479659882388, 5.66828971538788621540513019141, 5.83606009752520577561577590139, 6.32804281240571736527096397442, 7.00816169936187185627896678697, 7.44603916369347235577420959500, 7.54488443915516510283392574198, 8.114061847201436409623374491730, 8.728600786314345023625280564125, 8.881024066759561667128452120861, 9.389777395671716201513469034411, 9.596125079566744108661686525046

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