Properties

Label 4-230e2-1.1-c1e2-0-0
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 + 4·10-s + 3·11-s − 3·12-s + 7·13-s − 2·14-s + 2·15-s + 5·16-s − 3·17-s + 7·19-s − 6·20-s − 21-s − 6·22-s + 2·23-s + 4·24-s + 3·25-s − 14·26-s − 2·27-s + 3·28-s + 6·29-s − 4·30-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 + 1.26·10-s + 0.904·11-s − 0.866·12-s + 1.94·13-s − 0.534·14-s + 0.516·15-s + 5/4·16-s − 0.727·17-s + 1.60·19-s − 1.34·20-s − 0.218·21-s − 1.27·22-s + 0.417·23-s + 0.816·24-s + 3/5·25-s − 2.74·26-s − 0.384·27-s + 0.566·28-s + 1.11·29-s − 0.730·30-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

\(L(1)\) \(\approx\) \(0.5890151903\)
\(L(\frac12)\) \(\approx\) \(0.5890151903\)
\(L(\frac{3}{2})\) 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$D_{4}$ \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 7 T + 33 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 3 T + 31 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 7 T + 45 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 46 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 - 7 T + 27 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 9 T + 97 T^{2} + 9 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 18 T + 154 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$D_{4}$ \( 1 + 18 T + 178 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 7 T + 87 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 3 T + 97 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 2 T - 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$D_{4}$ \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 7 T + 75 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
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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

−11.94832209523499778169031549806, −11.74263410731868075600580313835, −11.41971376748791719919366959666, −11.14599489474051331468765485725, −10.51599717129527065742264627948, −10.10507068480568508870022571273, −9.492177683416520399108476133324, −8.891536565685023601694878685968, −8.501411878293945551716222971506, −8.230672694016464888617220280527, −7.57143152749856172316050467033, −6.97783199968102984923090359656, −6.41055578761420862138865627363, −6.20537997836224001827150418590, −5.18416600633045649339831766942, −4.60961578868387518096880916533, −3.51415309981038613162094802652, −3.25470319294818186689101430882, −1.69968985125155088025702730602, −0.911793692957645258149635391835, 0.911793692957645258149635391835, 1.69968985125155088025702730602, 3.25470319294818186689101430882, 3.51415309981038613162094802652, 4.60961578868387518096880916533, 5.18416600633045649339831766942, 6.20537997836224001827150418590, 6.41055578761420862138865627363, 6.97783199968102984923090359656, 7.57143152749856172316050467033, 8.230672694016464888617220280527, 8.501411878293945551716222971506, 8.891536565685023601694878685968, 9.492177683416520399108476133324, 10.10507068480568508870022571273, 10.51599717129527065742264627948, 11.14599489474051331468765485725, 11.41971376748791719919366959666, 11.74263410731868075600580313835, 11.94832209523499778169031549806

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