# Properties

 Label 4-1280e2-1.1-c0e2-0-5 Degree $4$ Conductor $1638400$ Sign $1$ Analytic cond. $0.408069$ Root an. cond. $0.799251$ Motivic weight $0$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s − 2·53-s + 4·61-s − 4·65-s + 2·73-s − 81-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + ⋯
 L(s)  = 1 + 2·5-s − 2·13-s + 2·17-s + 3·25-s − 2·37-s − 2·53-s + 4·61-s − 4·65-s + 2·73-s − 81-s + 4·85-s − 2·97-s − 2·113-s − 2·121-s + 4·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s + 179-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$1638400$$    =    $$2^{16} \cdot 5^{2}$$ Sign: $1$ Analytic conductor: $$0.408069$$ Root analytic conductor: $$0.799251$$ Motivic weight: $$0$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 1638400,\ (\ :0, 0),\ 1)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.524494646$$ $$L(\frac12)$$ $$\approx$$ $$1.524494646$$ $$L(1)$$ 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 $$1$$
5$C_1$ $$( 1 - T )^{2}$$
good3$C_2^2$ $$1 + T^{4}$$
7$C_2^2$ $$1 + T^{4}$$
11$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
17$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
19$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
23$C_2^2$ $$1 + T^{4}$$
29$C_2$ $$( 1 + T^{2} )^{2}$$
31$C_2$ $$( 1 + T^{2} )^{2}$$
37$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
41$C_2$ $$( 1 + T^{2} )^{2}$$
43$C_2^2$ $$1 + T^{4}$$
47$C_2^2$ $$1 + T^{4}$$
53$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
59$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
61$C_1$ $$( 1 - T )^{4}$$
67$C_2^2$ $$1 + T^{4}$$
71$C_2$ $$( 1 + T^{2} )^{2}$$
73$C_1$$\times$$C_2$ $$( 1 - T )^{2}( 1 + T^{2} )$$
79$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
83$C_2^2$ $$1 + T^{4}$$
89$C_1$$\times$$C_1$ $$( 1 - T )^{2}( 1 + T )^{2}$$
97$C_1$$\times$$C_2$ $$( 1 + T )^{2}( 1 + T^{2} )$$
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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

−9.964832605681071972461891305552, −9.799743619951564699777668367968, −9.291224714006104767552104887300, −9.186141973908284260333454055670, −8.291711220449639671757808819552, −8.248164377289671937300698850356, −7.52536394127897905866697190257, −7.17438654156441139716822791752, −6.59235801628521209496711360420, −6.57314167631080201188453586388, −5.70376606193782252549884416415, −5.41866129202010967020773057736, −5.05250576282773688095770727869, −4.99125058016544368812370535309, −3.96212254881160499698511916202, −3.43923422826357538745900098234, −2.70701320733826131018143723953, −2.48842071831408910394253778505, −1.77841542129944846033303856091, −1.20109791377981409238931281914, 1.20109791377981409238931281914, 1.77841542129944846033303856091, 2.48842071831408910394253778505, 2.70701320733826131018143723953, 3.43923422826357538745900098234, 3.96212254881160499698511916202, 4.99125058016544368812370535309, 5.05250576282773688095770727869, 5.41866129202010967020773057736, 5.70376606193782252549884416415, 6.57314167631080201188453586388, 6.59235801628521209496711360420, 7.17438654156441139716822791752, 7.52536394127897905866697190257, 8.248164377289671937300698850356, 8.291711220449639671757808819552, 9.186141973908284260333454055670, 9.291224714006104767552104887300, 9.799743619951564699777668367968, 9.964832605681071972461891305552