# Properties

 Degree $4$ Conductor $583200$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

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## 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