Properties

Label 4-10130-1.1-c1e2-0-0
Degree $4$
Conductor $10130$
Sign $-1$
Analytic cond. $0.645897$
Root an. cond. $0.896480$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s + 2·5-s − 4·7-s + 8-s − 4·9-s − 2·10-s − 11-s − 7·13-s + 4·14-s + 3·16-s + 4·18-s − 6·19-s − 2·20-s + 22-s + 4·23-s + 2·25-s + 7·26-s + 4·28-s + 3·29-s − 3·31-s − 3·32-s − 8·35-s + 4·36-s + 3·37-s + 6·38-s + 2·40-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.894·5-s − 1.51·7-s + 0.353·8-s − 4/3·9-s − 0.632·10-s − 0.301·11-s − 1.94·13-s + 1.06·14-s + 3/4·16-s + 0.942·18-s − 1.37·19-s − 0.447·20-s + 0.213·22-s + 0.834·23-s + 2/5·25-s + 1.37·26-s + 0.755·28-s + 0.557·29-s − 0.538·31-s − 0.530·32-s − 1.35·35-s + 2/3·36-s + 0.493·37-s + 0.973·38-s + 0.316·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10130 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10130 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10130\)    =    \(2 \cdot 5 \cdot 1013\)
Sign: $-1$
Analytic conductor: \(0.645897\)
Root analytic conductor: \(0.896480\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{10130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 10130,\ (\ :1/2, 1/2),\ -1)\)

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$$\times$$C_2$ \( ( 1 + T )( 1 + p T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 + T )( 1 - 3 T + p T^{2} ) \)
1013$C_1$$\times$$C_2$ \( ( 1 - T )( 1 - 35 T + p T^{2} ) \)
good3$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
7$C_2$$\times$$C_2$ \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \)
11$D_{4}$ \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 7 T + 33 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 22 T^{2} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 16 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 3 T + 3 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 3 T + 54 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 8 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 5 T + 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 13 T + 121 T^{2} - 13 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 7 T + 100 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 3 T + 9 T^{2} + 3 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 11 T + 67 T^{2} + 11 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 50 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 12 T + 112 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
79$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
83$D_{4}$ \( 1 - 7 T - 6 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + T - 99 T^{2} + p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 20 T + 240 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
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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

−16.9705009657, −16.6823097671, −16.0145032545, −15.2492916429, −14.8390964063, −14.2293951292, −14.0660916490, −13.1126867682, −12.9189322405, −12.4021606525, −11.9419072691, −10.9039897235, −10.5272409215, −9.96395120285, −9.38987967442, −9.17632518264, −8.67046031761, −7.81724768831, −7.22763325435, −6.31659580982, −5.92738277418, −5.21223927154, −4.34714388484, −2.99824975312, −2.54695632700, 0, 2.54695632700, 2.99824975312, 4.34714388484, 5.21223927154, 5.92738277418, 6.31659580982, 7.22763325435, 7.81724768831, 8.67046031761, 9.17632518264, 9.38987967442, 9.96395120285, 10.5272409215, 10.9039897235, 11.9419072691, 12.4021606525, 12.9189322405, 13.1126867682, 14.0660916490, 14.2293951292, 14.8390964063, 15.2492916429, 16.0145032545, 16.6823097671, 16.9705009657

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