Properties

Degree $4$
Conductor $5760$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s − 2·5-s + 6-s + 8-s − 2·9-s − 2·10-s + 12-s − 2·15-s + 16-s − 2·18-s − 5·19-s − 2·20-s + 3·23-s + 24-s + 2·25-s − 5·27-s − 3·29-s − 2·30-s + 32-s − 2·36-s − 5·38-s − 2·40-s + 7·43-s + 4·45-s + 3·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.353·8-s − 2/3·9-s − 0.632·10-s + 0.288·12-s − 0.516·15-s + 1/4·16-s − 0.471·18-s − 1.14·19-s − 0.447·20-s + 0.625·23-s + 0.204·24-s + 2/5·25-s − 0.962·27-s − 0.557·29-s − 0.365·30-s + 0.176·32-s − 1/3·36-s − 0.811·38-s − 0.316·40-s + 1.06·43-s + 0.596·45-s + 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5760 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5760\)    =    \(2^{7} \cdot 3^{2} \cdot 5\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{5760} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5760,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.255225901\)
\(L(\frac12)\) \(\approx\) \(1.255225901\)
\(L(\frac{3}{2})\) 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$C_1$ \( 1 - T \)
3$C_2$ \( 1 - T + p T^{2} \)
5$C_1$$\times$$C_2$ \( ( 1 - T )( 1 + 3 T + p T^{2} ) \)
good7$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T^{2} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 19 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
23$C_2$$\times$$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
43$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
47$C_2$$\times$$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 - 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 80 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2^2$ \( 1 + 103 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 - 113 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 43 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + T + p T^{2} ) \)
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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

−12.24244473605294519572636046830, −11.51133021956550516374153915983, −11.00094382278444183710937467646, −10.72395766258735584990921711927, −9.679519757957213208192705975936, −9.038534843497895978589661319811, −8.461073727372260995859162050171, −7.83940290047431122739056812391, −7.31686954518840004795349045416, −6.46214012792232145958811426138, −5.77741860069763357998984744376, −4.85658574499485116494790804675, −4.05913985275459354094038282359, −3.33988096585002869805307011926, −2.36023487352734216619085451403, 2.36023487352734216619085451403, 3.33988096585002869805307011926, 4.05913985275459354094038282359, 4.85658574499485116494790804675, 5.77741860069763357998984744376, 6.46214012792232145958811426138, 7.31686954518840004795349045416, 7.83940290047431122739056812391, 8.461073727372260995859162050171, 9.038534843497895978589661319811, 9.679519757957213208192705975936, 10.72395766258735584990921711927, 11.00094382278444183710937467646, 11.51133021956550516374153915983, 12.24244473605294519572636046830

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