Properties

Label 4-200e2-1.1-c1e2-0-4
Degree $4$
Conductor $40000$
Sign $1$
Analytic cond. $2.55043$
Root an. cond. $1.26372$
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  + 2-s − 2·3-s + 4-s − 2·6-s + 8-s − 3·9-s + 2·11-s − 2·12-s + 4·13-s + 16-s − 2·17-s − 3·18-s + 10·19-s + 2·22-s − 12·23-s − 2·24-s + 4·26-s + 14·27-s + 4·29-s + 12·31-s + 32-s − 4·33-s − 2·34-s − 3·36-s + 8·37-s + 10·38-s − 8·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s − 9-s + 0.603·11-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 2.29·19-s + 0.426·22-s − 2.50·23-s − 0.408·24-s + 0.784·26-s + 2.69·27-s + 0.742·29-s + 2.15·31-s + 0.176·32-s − 0.696·33-s − 0.342·34-s − 1/2·36-s + 1.31·37-s + 1.62·38-s − 1.28·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(40000\)    =    \(2^{6} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(2.55043\)
Root analytic conductor: \(1.26372\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 40000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.329953899\)
\(L(\frac12)\) \(\approx\) \(1.329953899\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
5 \( 1 \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
13$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
17$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
29$C_2$$\times$$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
53$C_2$$\times$$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \)
89$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
97$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
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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

−14.8940629672, −14.2754339531, −14.0249630004, −13.7407879771, −13.3181822032, −12.3320214092, −12.1846203232, −11.6436611983, −11.4154247282, −11.1600742237, −10.3542554150, −9.93072373032, −9.37569265672, −8.59823467768, −8.07172921856, −7.75497054910, −6.54551923808, −6.35306147257, −6.01711432754, −5.34945783065, −4.83783296054, −4.05979688076, −3.29699975126, −2.56583690538, −1.05700989626, 1.05700989626, 2.56583690538, 3.29699975126, 4.05979688076, 4.83783296054, 5.34945783065, 6.01711432754, 6.35306147257, 6.54551923808, 7.75497054910, 8.07172921856, 8.59823467768, 9.37569265672, 9.93072373032, 10.3542554150, 11.1600742237, 11.4154247282, 11.6436611983, 12.1846203232, 12.3320214092, 13.3181822032, 13.7407879771, 14.0249630004, 14.2754339531, 14.8940629672

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