Properties

Degree $4$
Conductor $532400$
Sign $1$
Motivic weight $1$
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 − 2·3-s + 2·4-s + 5-s − 4·6-s − 3·9-s + 2·10-s − 11-s − 4·12-s − 8·13-s − 2·15-s − 4·16-s + 4·17-s − 6·18-s + 2·20-s − 2·22-s − 2·23-s − 4·25-s − 16·26-s + 14·27-s − 4·30-s − 8·32-s + 2·33-s + 8·34-s − 6·36-s + 16·39-s − 2·44-s + ⋯
L(s)  = 1  + 1.41·2-s − 1.15·3-s + 4-s + 0.447·5-s − 1.63·6-s − 9-s + 0.632·10-s − 0.301·11-s − 1.15·12-s − 2.21·13-s − 0.516·15-s − 16-s + 0.970·17-s − 1.41·18-s + 0.447·20-s − 0.426·22-s − 0.417·23-s − 4/5·25-s − 3.13·26-s + 2.69·27-s − 0.730·30-s − 1.41·32-s + 0.348·33-s + 1.37·34-s − 36-s + 2.56·39-s − 0.301·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 532400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(532400\)    =    \(2^{4} \cdot 5^{2} \cdot 11^{3}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{532400} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 532400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.147836550\)
\(L(\frac12)\) \(\approx\) \(1.147836550\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - p T + p T^{2} \)
5$C_2$ \( 1 - T + p T^{2} \)
11$C_1$ \( 1 + T \)
good3$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
7$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
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   \(L(s) = \displaystyle\prod_{\mathfrak{p}\ \mathrm{bad}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s})^{-1} \prod_{\mathfrak{p}\ \mathrm{good}} (1- a(\mathfrak{p}) (N\mathfrak{p})^{-s} + (N\mathfrak{p})^{-2s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.394238320905654147490500873996, −7.78789607043645765017922373874, −7.49534540311409429819739808064, −6.89747784534263149853829967422, −6.23870064789214498217427813253, −6.10845204313762133714030223921, −5.41175071356617719099292784896, −5.40055166000655653433368694834, −4.88140290332121063087048287423, −4.44574490854809981150981932393, −3.68709007429289538959412107978, −2.96905370396904801865789065276, −2.63044898935838963010520851422, −2.00337434233232539103171654262, −0.44102589544325217691153494662, 0.44102589544325217691153494662, 2.00337434233232539103171654262, 2.63044898935838963010520851422, 2.96905370396904801865789065276, 3.68709007429289538959412107978, 4.44574490854809981150981932393, 4.88140290332121063087048287423, 5.40055166000655653433368694834, 5.41175071356617719099292784896, 6.10845204313762133714030223921, 6.23870064789214498217427813253, 6.89747784534263149853829967422, 7.49534540311409429819739808064, 7.78789607043645765017922373874, 8.394238320905654147490500873996

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