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Properties

Label 4-77e2-1.1-c1e2-0-2
Degree $4$
Conductor $5929$
Sign $1$
Analytic cond. $0.378038$
Root an. cond. $0.784122$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 4-s − 4·5-s + 2·7-s + 2·9-s − 2·11-s + 2·12-s + 2·13-s − 8·15-s − 3·16-s − 2·17-s + 4·19-s − 4·20-s + 4·21-s − 4·23-s + 2·25-s + 6·27-s + 2·28-s + 8·29-s − 10·31-s − 4·33-s − 8·35-s + 2·36-s − 8·37-s + 4·39-s − 18·41-s + 16·43-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/2·4-s − 1.78·5-s + 0.755·7-s + 2/3·9-s − 0.603·11-s + 0.577·12-s + 0.554·13-s − 2.06·15-s − 3/4·16-s − 0.485·17-s + 0.917·19-s − 0.894·20-s + 0.872·21-s − 0.834·23-s + 2/5·25-s + 1.15·27-s + 0.377·28-s + 1.48·29-s − 1.79·31-s − 0.696·33-s − 1.35·35-s + 1/3·36-s − 1.31·37-s + 0.640·39-s − 2.81·41-s + 2.43·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5929\)    =    \(7^{2} \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(0.378038\)
Root analytic conductor: \(0.784122\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5929,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.070586424\)
\(L(\frac12)\) \(\approx\) \(1.070586424\)
\(L(\frac{3}{2})\) 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)$
bad7$C_1$ \( ( 1 - T )^{2} \)
11$C_1$ \( ( 1 + T )^{2} \)
good2$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
5$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$D_{4}$ \( 1 - 2 T + 22 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$C_4$ \( 1 - 8 T + 54 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 82 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 - 10 T + 114 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 2 T + 114 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 10 T + 142 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 20 T + 214 T^{2} - 20 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 6 T + 150 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 78 T^{2} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 8 T + 30 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
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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

−14.77113383902928186552317166788, −14.27550916790215652419511200225, −13.64378596260725423988467239504, −13.49781163498927232710566477325, −12.18910384179657484724636644400, −12.18782379390857727025883054502, −11.59981959396268078086031221575, −10.80098618032781986788043378165, −10.66696977835995280693652080171, −9.646008721917093568598530969159, −8.655694758854538486237566014906, −8.578069454461321602627873782574, −7.921452813266244493658886433521, −7.29080846889334209401330189578, −7.01529740975175196472359515240, −5.72112728486865574624740022438, −4.69824918973277729331348697866, −3.96714211787804829284362244522, −3.28451169009027979961022861738, −2.16581206833890556742345562501, 2.16581206833890556742345562501, 3.28451169009027979961022861738, 3.96714211787804829284362244522, 4.69824918973277729331348697866, 5.72112728486865574624740022438, 7.01529740975175196472359515240, 7.29080846889334209401330189578, 7.921452813266244493658886433521, 8.578069454461321602627873782574, 8.655694758854538486237566014906, 9.646008721917093568598530969159, 10.66696977835995280693652080171, 10.80098618032781986788043378165, 11.59981959396268078086031221575, 12.18782379390857727025883054502, 12.18910384179657484724636644400, 13.49781163498927232710566477325, 13.64378596260725423988467239504, 14.27550916790215652419511200225, 14.77113383902928186552317166788

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