Properties

Label 4-600e2-1.1-c1e2-0-36
Degree $4$
Conductor $360000$
Sign $-1$
Analytic cond. $22.9539$
Root an. cond. $2.18884$
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 + 2·3-s − 4-s + 2·6-s − 3·8-s + 3·9-s − 8·11-s − 2·12-s − 16-s − 4·17-s + 3·18-s + 8·19-s − 8·22-s − 6·24-s + 4·27-s + 5·32-s − 16·33-s − 4·34-s − 3·36-s + 8·38-s + 20·41-s − 8·43-s + 8·44-s − 2·48-s − 14·49-s − 8·51-s + 4·54-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s − 1/2·4-s + 0.816·6-s − 1.06·8-s + 9-s − 2.41·11-s − 0.577·12-s − 1/4·16-s − 0.970·17-s + 0.707·18-s + 1.83·19-s − 1.70·22-s − 1.22·24-s + 0.769·27-s + 0.883·32-s − 2.78·33-s − 0.685·34-s − 1/2·36-s + 1.29·38-s + 3.12·41-s − 1.21·43-s + 1.20·44-s − 0.288·48-s − 2·49-s − 1.12·51-s + 0.544·54-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(360000\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{4}\)
Sign: $-1$
Analytic conductor: \(22.9539\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{360000} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 360000,\ (\ :1/2, 1/2),\ -1)\)

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

−8.603932276417102120989834189554, −7.84465462281701093084420479537, −7.68003844385765985083943135134, −7.39546104131459972836674622338, −6.60942929553535216154216963636, −5.86497653842965752491328535183, −5.59929057414059862389948014289, −4.97287019204270070424993179755, −4.52160444884874669934496061646, −4.17881549969972779526027182436, −3.10526976657196724130917180316, −3.01549784140686353642765162463, −2.58465419848810939203642390782, −1.51207711394055808445468187309, 0, 1.51207711394055808445468187309, 2.58465419848810939203642390782, 3.01549784140686353642765162463, 3.10526976657196724130917180316, 4.17881549969972779526027182436, 4.52160444884874669934496061646, 4.97287019204270070424993179755, 5.59929057414059862389948014289, 5.86497653842965752491328535183, 6.60942929553535216154216963636, 7.39546104131459972836674622338, 7.68003844385765985083943135134, 7.84465462281701093084420479537, 8.603932276417102120989834189554

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