Properties

Label 4-300e2-1.1-c0e2-0-0
Degree $4$
Conductor $90000$
Sign $1$
Analytic cond. $0.0224159$
Root an. cond. $0.386936$
Motivic weight $0$
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-s + 2·19-s − 2·31-s + 49-s − 2·61-s − 4·79-s + 81-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯
L(s)  = 1  − 9-s + 2·19-s − 2·31-s + 49-s − 2·61-s − 4·79-s + 81-s + 2·109-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 2·171-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(90000\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(0.0224159\)
Root analytic conductor: \(0.386936\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{300} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 90000,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5760602672\)
\(L(\frac12)\) \(\approx\) \(0.5760602672\)
\(L(1)\) 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$C_2$ \( 1 + T^{2} \)
5 \( 1 \)
good7$C_2^2$ \( 1 - T^{2} + T^{4} \)
11$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
13$C_2^2$ \( 1 - T^{2} + T^{4} \)
17$C_2$ \( ( 1 + T^{2} )^{2} \)
19$C_2$ \( ( 1 - T + T^{2} )^{2} \)
23$C_2$ \( ( 1 + T^{2} )^{2} \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 + T + T^{2} )^{2} \)
37$C_2$ \( ( 1 + T^{2} )^{2} \)
41$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
43$C_2^2$ \( 1 - T^{2} + T^{4} \)
47$C_2$ \( ( 1 + T^{2} )^{2} \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
61$C_2$ \( ( 1 + T + T^{2} )^{2} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 + T^{2} )^{2} \)
79$C_1$ \( ( 1 + T )^{4} \)
83$C_2$ \( ( 1 + T^{2} )^{2} \)
89$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
97$C_2^2$ \( 1 - T^{2} + T^{4} \)
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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

−12.01372016912809342749285172174, −11.78061059495766263985688721443, −11.12320549034651232018049959416, −11.11616451565845491899036330357, −10.24535189912948783281184977550, −9.929993887286861233702550480290, −9.302592474543976489563525055459, −8.873022695828254400712013405300, −8.611554033917077913091286940942, −7.72608082543691412317778600171, −7.44423218924716347098249162256, −7.05929133007431612511485815057, −6.10915812908886137955795875640, −5.76908640453884171589561568387, −5.29423475529553211433386184685, −4.69009675734075972868552749914, −3.80587507684635087489723777263, −3.23349964907802010162953016143, −2.63549969015813420091347628242, −1.51565973822216203176734940741, 1.51565973822216203176734940741, 2.63549969015813420091347628242, 3.23349964907802010162953016143, 3.80587507684635087489723777263, 4.69009675734075972868552749914, 5.29423475529553211433386184685, 5.76908640453884171589561568387, 6.10915812908886137955795875640, 7.05929133007431612511485815057, 7.44423218924716347098249162256, 7.72608082543691412317778600171, 8.611554033917077913091286940942, 8.873022695828254400712013405300, 9.302592474543976489563525055459, 9.929993887286861233702550480290, 10.24535189912948783281184977550, 11.11616451565845491899036330357, 11.12320549034651232018049959416, 11.78061059495766263985688721443, 12.01372016912809342749285172174

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