Properties

Degree $4$
Conductor $583200$
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-s + 4-s + 3·5-s − 2·7-s − 8-s − 3·10-s − 6·11-s + 2·14-s + 16-s + 3·20-s + 6·22-s + 4·25-s − 2·28-s − 32-s − 6·35-s − 3·40-s − 20·43-s − 6·44-s − 11·49-s − 4·50-s + 18·53-s − 18·55-s + 2·56-s + 24·59-s + 16·61-s + 64-s + 28·67-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.755·7-s − 0.353·8-s − 0.948·10-s − 1.80·11-s + 0.534·14-s + 1/4·16-s + 0.670·20-s + 1.27·22-s + 4/5·25-s − 0.377·28-s − 0.176·32-s − 1.01·35-s − 0.474·40-s − 3.04·43-s − 0.904·44-s − 1.57·49-s − 0.565·50-s + 2.47·53-s − 2.42·55-s + 0.267·56-s + 3.12·59-s + 2.04·61-s + 1/8·64-s + 3.42·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 583200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(583200\)    =    \(2^{5} \cdot 3^{6} \cdot 5^{2}\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{583200} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 583200,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.143786255\)
\(L(\frac12)\) \(\approx\) \(1.143786255\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5$C_2$ \( 1 - 3 T + p T^{2} \)
good7$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
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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

−8.310249526893034803263699830740, −8.268048909391662018895990050852, −7.58907033734171805431789545424, −6.84593942818273157797656705828, −6.73860900674537002760989556093, −6.35572304303649648480081327455, −5.38354729541915242732826157737, −5.37363015537505503550836847886, −5.13499928761574618124626716865, −3.92861197755400275794059610293, −3.48774792613401599337251547075, −2.58391653549265790674050070138, −2.44850978333263328730546301919, −1.72580411383323342691331766149, −0.58944949786391246601175874957, 0.58944949786391246601175874957, 1.72580411383323342691331766149, 2.44850978333263328730546301919, 2.58391653549265790674050070138, 3.48774792613401599337251547075, 3.92861197755400275794059610293, 5.13499928761574618124626716865, 5.37363015537505503550836847886, 5.38354729541915242732826157737, 6.35572304303649648480081327455, 6.73860900674537002760989556093, 6.84593942818273157797656705828, 7.58907033734171805431789545424, 8.268048909391662018895990050852, 8.310249526893034803263699830740

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