# Properties

 Label 4-280e2-1.1-c0e2-0-0 Degree $4$ Conductor $78400$ Sign $1$ Analytic cond. $0.0195267$ Root an. cond. $0.373815$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2-s − 5-s + 2·7-s + 8-s − 9-s + 10-s + 11-s − 2·13-s − 2·14-s − 16-s + 18-s + 19-s − 22-s + 23-s + 2·26-s − 2·35-s + 37-s − 38-s − 40-s − 2·41-s + 45-s − 46-s + 47-s + 3·49-s + 53-s − 55-s + 2·56-s + ⋯
 L(s)  = 1 − 2-s − 5-s + 2·7-s + 8-s − 9-s + 10-s + 11-s − 2·13-s − 2·14-s − 16-s + 18-s + 19-s − 22-s + 23-s + 2·26-s − 2·35-s + 37-s − 38-s − 40-s − 2·41-s + 45-s − 46-s + 47-s + 3·49-s + 53-s − 55-s + 2·56-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$78400$$    =    $$2^{6} \cdot 5^{2} \cdot 7^{2}$$ Sign: $1$ Analytic conductor: $$0.0195267$$ Root analytic conductor: $$0.373815$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{280} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 78400,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.3243421534$$ $$L(\frac12)$$ $$\approx$$ $$0.3243421534$$ $$L(1)$$ 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$C_2$ $$1 + T + T^{2}$$
5$C_2$ $$1 + T + T^{2}$$
7$C_1$ $$( 1 - T )^{2}$$
good3$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
11$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
13$C_2$ $$( 1 + T + T^{2} )^{2}$$
17$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
19$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
23$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
29$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
31$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
37$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
41$C_2$ $$( 1 + T + T^{2} )^{2}$$
43$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
47$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
53$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T + T^{2} )$$
59$C_2$ $$( 1 + T + T^{2} )^{2}$$
61$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
67$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
71$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
73$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
79$C_2$ $$( 1 - T + T^{2} )( 1 + T + T^{2} )$$
83$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
89$C_2$ $$( 1 + T + T^{2} )^{2}$$
97$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
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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

−12.13830326851865339363536546666, −11.80071508954180853926845378276, −11.20615371565957059761013023842, −11.13932298076329041515698727402, −10.48062927226413804045761639682, −9.860082458716909779214524538077, −9.368086750882871791009580674703, −8.973991931372176462136622755714, −8.292953158098930586523783564286, −8.244697824609083528068906561575, −7.45877146856009408561287802246, −7.38314043813070187558221167778, −6.79736537383575133286069677998, −5.53551928462064910010219100853, −5.26808382279490391258148059126, −4.45199213217626417055744200907, −4.38038338526091381188467776575, −3.23446085663219780964219393327, −2.30247443887687932688881450063, −1.29490937665423030484310327694, 1.29490937665423030484310327694, 2.30247443887687932688881450063, 3.23446085663219780964219393327, 4.38038338526091381188467776575, 4.45199213217626417055744200907, 5.26808382279490391258148059126, 5.53551928462064910010219100853, 6.79736537383575133286069677998, 7.38314043813070187558221167778, 7.45877146856009408561287802246, 8.244697824609083528068906561575, 8.292953158098930586523783564286, 8.973991931372176462136622755714, 9.368086750882871791009580674703, 9.860082458716909779214524538077, 10.48062927226413804045761639682, 11.13932298076329041515698727402, 11.20615371565957059761013023842, 11.80071508954180853926845378276, 12.13830326851865339363536546666