Properties

Label 4-540000-1.1-c1e2-0-9
Degree $4$
Conductor $540000$
Sign $-1$
Analytic cond. $34.4308$
Root an. cond. $2.42235$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 4·7-s − 8-s + 9-s − 4·11-s + 12-s + 4·14-s + 16-s + 4·17-s − 18-s − 4·21-s + 4·22-s − 24-s + 27-s − 4·28-s − 32-s − 4·33-s − 4·34-s + 36-s + 4·42-s + 8·43-s − 4·44-s + 48-s − 2·49-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 1.06·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s − 0.872·21-s + 0.852·22-s − 0.204·24-s + 0.192·27-s − 0.755·28-s − 0.176·32-s − 0.696·33-s − 0.685·34-s + 1/6·36-s + 0.617·42-s + 1.21·43-s − 0.603·44-s + 0.144·48-s − 2/7·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(540000\)    =    \(2^{5} \cdot 3^{3} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(34.4308\)
Root analytic conductor: \(2.42235\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{540000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 540000,\ (\ :1/2, 1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 + T \)
3$C_1$ \( 1 - T \)
5 \( 1 \)
good7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
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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

−8.236259450125909594320955817325, −7.79436266718632723136861068704, −7.34830902118093822137916374938, −7.19091277442794332068165955627, −6.39955394184406631525545593399, −6.04674329942688851458805581356, −5.66765515312328094809085573246, −4.95720185791601620138850804643, −4.40421151945420287303938949967, −3.43332661175523197849238647019, −3.36984414779728341388392819787, −2.66426897593599555100997020524, −2.17693746121355235992569926706, −1.09380620717066232730427473522, 0, 1.09380620717066232730427473522, 2.17693746121355235992569926706, 2.66426897593599555100997020524, 3.36984414779728341388392819787, 3.43332661175523197849238647019, 4.40421151945420287303938949967, 4.95720185791601620138850804643, 5.66765515312328094809085573246, 6.04674329942688851458805581356, 6.39955394184406631525545593399, 7.19091277442794332068165955627, 7.34830902118093822137916374938, 7.79436266718632723136861068704, 8.236259450125909594320955817325

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