Properties

Label 4-2000-1.1-c1e2-0-0
Degree $4$
Conductor $2000$
Sign $1$
Analytic cond. $0.127521$
Root an. cond. $0.597579$
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  − 5-s − 2·9-s − 8·19-s + 25-s + 12·29-s − 8·31-s + 12·41-s + 2·45-s − 10·49-s + 24·59-s + 4·61-s − 24·71-s + 16·79-s − 5·81-s − 12·89-s + 8·95-s + 12·101-s + 4·109-s − 22·121-s − 125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + ⋯
L(s)  = 1  − 0.447·5-s − 2/3·9-s − 1.83·19-s + 1/5·25-s + 2.22·29-s − 1.43·31-s + 1.87·41-s + 0.298·45-s − 1.42·49-s + 3.12·59-s + 0.512·61-s − 2.84·71-s + 1.80·79-s − 5/9·81-s − 1.27·89-s + 0.820·95-s + 1.19·101-s + 0.383·109-s − 2·121-s − 0.0894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2000\)    =    \(2^{4} \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(0.127521\)
Root analytic conductor: \(0.597579\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5945775518\)
\(L(\frac12)\) \(\approx\) \(0.5945775518\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
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

−13.00236641577412261230757825497, −12.92148359796847673035103596896, −12.03947269145466851839351538613, −11.50854453196989171707939391081, −10.83473950301065015207539180976, −10.36534498208386853525755121224, −9.493169657561115216422783286207, −8.552217550781204179646223792184, −8.414042470938792927705837581638, −7.38409000222510611898226618037, −6.57891116465648258947670054106, −5.85740317846722582719715899176, −4.78130792717525308450176413839, −3.90074996505307731038192172560, −2.58321256178540655780606724063, 2.58321256178540655780606724063, 3.90074996505307731038192172560, 4.78130792717525308450176413839, 5.85740317846722582719715899176, 6.57891116465648258947670054106, 7.38409000222510611898226618037, 8.414042470938792927705837581638, 8.552217550781204179646223792184, 9.493169657561115216422783286207, 10.36534498208386853525755121224, 10.83473950301065015207539180976, 11.50854453196989171707939391081, 12.03947269145466851839351538613, 12.92148359796847673035103596896, 13.00236641577412261230757825497

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