Properties

Degree $4$
Conductor $521284$
Sign $1$
Motivic weight $5$
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s − 3·3-s + 48·4-s − 45·5-s − 24·6-s + 114·7-s + 256·8-s − 119·9-s − 360·10-s + 661·11-s − 144·12-s − 1.61e3·13-s + 912·14-s + 135·15-s + 1.28e3·16-s + 64·17-s − 952·18-s − 2.16e3·20-s − 342·21-s + 5.28e3·22-s − 3.18e3·23-s − 768·24-s − 1.48e3·25-s − 1.29e4·26-s + 12·27-s + 5.47e3·28-s + 2.48e3·29-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.192·3-s + 3/2·4-s − 0.804·5-s − 0.272·6-s + 0.879·7-s + 1.41·8-s − 0.489·9-s − 1.13·10-s + 1.64·11-s − 0.288·12-s − 2.64·13-s + 1.24·14-s + 0.154·15-s + 5/4·16-s + 0.0537·17-s − 0.692·18-s − 1.20·20-s − 0.169·21-s + 2.32·22-s − 1.25·23-s − 0.272·24-s − 0.476·25-s − 3.74·26-s + 0.00316·27-s + 1.31·28-s + 0.547·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 521284 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(521284\)    =    \(2^{2} \cdot 19^{4}\)
Sign: $1$
Motivic weight: \(5\)
Character: induced by $\chi_{722} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 521284,\ (\ :5/2, 5/2),\ 1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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$C_1$ \( ( 1 - p^{2} T )^{2} \)
19 \( 1 \)
good3$D_{4}$ \( 1 + p T + 128 T^{2} + p^{6} T^{3} + p^{10} T^{4} \)
5$D_{4}$ \( 1 + 9 p T + 3514 T^{2} + 9 p^{6} T^{3} + p^{10} T^{4} \)
7$D_{4}$ \( 1 - 114 T + 31099 T^{2} - 114 p^{5} T^{3} + p^{10} T^{4} \)
11$D_{4}$ \( 1 - 661 T + 430972 T^{2} - 661 p^{5} T^{3} + p^{10} T^{4} \)
13$D_{4}$ \( 1 + 1613 T + 1384022 T^{2} + 1613 p^{5} T^{3} + p^{10} T^{4} \)
17$D_{4}$ \( 1 - 64 T + 2804713 T^{2} - 64 p^{5} T^{3} + p^{10} T^{4} \)
23$D_{4}$ \( 1 + 3185 T + 14543782 T^{2} + 3185 p^{5} T^{3} + p^{10} T^{4} \)
29$D_{4}$ \( 1 - 2481 T + 6815332 T^{2} - 2481 p^{5} T^{3} + p^{10} T^{4} \)
31$D_{4}$ \( 1 - 1180 T + 21633278 T^{2} - 1180 p^{5} T^{3} + p^{10} T^{4} \)
37$D_{4}$ \( 1 + 10488 T + 155529814 T^{2} + 10488 p^{5} T^{3} + p^{10} T^{4} \)
41$D_{4}$ \( 1 + 16630 T + 170295586 T^{2} + 16630 p^{5} T^{3} + p^{10} T^{4} \)
43$D_{4}$ \( 1 - 11303 T + 297510638 T^{2} - 11303 p^{5} T^{3} + p^{10} T^{4} \)
47$D_{4}$ \( 1 + 12155 T + 47754214 T^{2} + 12155 p^{5} T^{3} + p^{10} T^{4} \)
53$D_{4}$ \( 1 + 20585 T + 812203882 T^{2} + 20585 p^{5} T^{3} + p^{10} T^{4} \)
59$D_{4}$ \( 1 - 78581 T + 2971672216 T^{2} - 78581 p^{5} T^{3} + p^{10} T^{4} \)
61$D_{4}$ \( 1 - 43621 T + 1919695356 T^{2} - 43621 p^{5} T^{3} + p^{10} T^{4} \)
67$D_{4}$ \( 1 + 7805 T - 748756690 T^{2} + 7805 p^{5} T^{3} + p^{10} T^{4} \)
71$D_{4}$ \( 1 - 62488 T + 4005427642 T^{2} - 62488 p^{5} T^{3} + p^{10} T^{4} \)
73$D_{4}$ \( 1 - 16218 T + 1004140843 T^{2} - 16218 p^{5} T^{3} + p^{10} T^{4} \)
79$D_{4}$ \( 1 + 67122 T + 7273984870 T^{2} + 67122 p^{5} T^{3} + p^{10} T^{4} \)
83$D_{4}$ \( 1 + 10714 T + 5751433246 T^{2} + 10714 p^{5} T^{3} + p^{10} T^{4} \)
89$D_{4}$ \( 1 + 128188 T + 8689285330 T^{2} + 128188 p^{5} T^{3} + p^{10} T^{4} \)
97$D_{4}$ \( 1 + 178558 T + 22668743394 T^{2} + 178558 p^{5} T^{3} + p^{10} T^{4} \)
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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

−9.639697000395356983664672259105, −8.972836679031130040945183979554, −8.232176697495258518185271667459, −8.110705677802465804289182662283, −7.67154200827645059263701850374, −6.90540272108608536310947990030, −6.83437592666392758265610783377, −6.47473839113900235586880530775, −5.51201723960271204536313721975, −5.31116791152855112130258181035, −4.95770561863875806426837089839, −4.34445457776065799719279065599, −3.87032186674611192676191852712, −3.76051539946298127985799676609, −2.70763265176811655994264136006, −2.49770640677928188718258621651, −1.70034429193977377944645637006, −1.29796450615623068183450847152, 0, 0, 1.29796450615623068183450847152, 1.70034429193977377944645637006, 2.49770640677928188718258621651, 2.70763265176811655994264136006, 3.76051539946298127985799676609, 3.87032186674611192676191852712, 4.34445457776065799719279065599, 4.95770561863875806426837089839, 5.31116791152855112130258181035, 5.51201723960271204536313721975, 6.47473839113900235586880530775, 6.83437592666392758265610783377, 6.90540272108608536310947990030, 7.67154200827645059263701850374, 8.110705677802465804289182662283, 8.232176697495258518185271667459, 8.972836679031130040945183979554, 9.639697000395356983664672259105

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