Properties

Label 4-230e2-1.1-c1e2-0-3
Degree $4$
Conductor $52900$
Sign $1$
Analytic cond. $3.37294$
Root an. cond. $1.35519$
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  + 2·2-s + 3-s + 3·4-s + 2·5-s + 2·6-s + 7-s + 4·8-s − 4·9-s + 4·10-s + 11-s + 3·12-s − 3·13-s + 2·14-s + 2·15-s + 5·16-s + 17-s − 8·18-s − 3·19-s + 6·20-s + 21-s + 2·22-s + 2·23-s + 4·24-s + 3·25-s − 6·26-s − 6·27-s + 3·28-s + ⋯
L(s)  = 1  + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s + 0.377·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s + 0.301·11-s + 0.866·12-s − 0.832·13-s + 0.534·14-s + 0.516·15-s + 5/4·16-s + 0.242·17-s − 1.88·18-s − 0.688·19-s + 1.34·20-s + 0.218·21-s + 0.426·22-s + 0.417·23-s + 0.816·24-s + 3/5·25-s − 1.17·26-s − 1.15·27-s + 0.566·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(52900\)    =    \(2^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(3.37294\)
Root analytic conductor: \(1.35519\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{230} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 52900,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.057957440\)
\(L(\frac12)\) \(\approx\) \(4.057957440\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
5$C_1$ \( ( 1 - T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good3$D_{4}$ \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 13 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + p T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 3 T - 3 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - T + 3 T^{2} - p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 3 T + 29 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 + 14 T + 102 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 43 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 9 T + p T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 6 T + 58 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 10 T + 98 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 3 T + 63 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 113 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 2 T + 142 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 12 T + 174 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 + 12 T + 194 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 27 T + 375 T^{2} - 27 p T^{3} + p^{2} T^{4} \)
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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

−12.47883434393512256328344394257, −12.15772750434465927058119918655, −11.44686251142789410021465049921, −11.19558779221915460508966203868, −10.82084895697821165088256119833, −10.01325729367604094589154766938, −9.542030720090398116833551036626, −9.140306759551471116968745802587, −8.377340443911295124746456879666, −7.996795640306081590736648146812, −7.37284920887802757557061253147, −6.70407123397807318145608994971, −6.06854457591781683225897468344, −5.81865497788505764214963526118, −4.89069048655957089784278056400, −4.88440935172606354314966556919, −3.67058546540356043216825506916, −3.21835418156143581887086605491, −2.38964411962337051053951412739, −1.89317783414266679508333114377, 1.89317783414266679508333114377, 2.38964411962337051053951412739, 3.21835418156143581887086605491, 3.67058546540356043216825506916, 4.88440935172606354314966556919, 4.89069048655957089784278056400, 5.81865497788505764214963526118, 6.06854457591781683225897468344, 6.70407123397807318145608994971, 7.37284920887802757557061253147, 7.996795640306081590736648146812, 8.377340443911295124746456879666, 9.140306759551471116968745802587, 9.542030720090398116833551036626, 10.01325729367604094589154766938, 10.82084895697821165088256119833, 11.19558779221915460508966203868, 11.44686251142789410021465049921, 12.15772750434465927058119918655, 12.47883434393512256328344394257

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