# Properties

 Degree $2$ Conductor $176400$ Sign $1$ Motivic weight $1$ Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + 2·11-s + 2·13-s + 4·17-s − 8·23-s − 2·31-s − 8·37-s − 2·41-s − 2·43-s + 10·47-s − 2·53-s − 4·59-s + 10·61-s + 2·67-s − 12·71-s + 10·73-s − 16·79-s + 16·83-s + 14·89-s + 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
 L(s)  = 1 + 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.66·23-s − 0.359·31-s − 1.31·37-s − 0.312·41-s − 0.304·43-s + 1.45·47-s − 0.274·53-s − 0.520·59-s + 1.28·61-s + 0.244·67-s − 1.42·71-s + 1.17·73-s − 1.80·79-s + 1.75·83-s + 1.48·89-s + 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$176400$$    =    $$2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}$$ Sign: $1$ Motivic weight: $$1$$ Character: $\chi_{176400} (1, \cdot )$ Sato-Tate group: $\mathrm{SU}(2)$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 176400,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.213356018$$ $$L(\frac12)$$ $$\approx$$ $$2.213356018$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
5 $$1$$
7 $$1$$
good11 $$1 - 2 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 4 T + p T^{2}$$
19 $$1 + p T^{2}$$
23 $$1 + 8 T + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 2 T + p T^{2}$$
37 $$1 + 8 T + p T^{2}$$
41 $$1 + 2 T + p T^{2}$$
43 $$1 + 2 T + p T^{2}$$
47 $$1 - 10 T + p T^{2}$$
53 $$1 + 2 T + p T^{2}$$
59 $$1 + 4 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 + 12 T + p T^{2}$$
73 $$1 - 10 T + p T^{2}$$
79 $$1 + 16 T + p T^{2}$$
83 $$1 - 16 T + p T^{2}$$
89 $$1 - 14 T + p T^{2}$$
97 $$1 - 6 T + p T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−13.22216015109653, −12.64660971198113, −12.11152099088996, −11.88064543471111, −11.42403267706662, −10.71616089464343, −10.28263640533346, −10.02758741544144, −9.244948603106116, −8.982002846163236, −8.390930751673256, −7.823308242891913, −7.552899562667763, −6.714204986471061, −6.480596833828254, −5.731102940099073, −5.482989704200729, −4.793316890450383, −4.015197241347150, −3.769911024965562, −3.213833168171274, −2.422841196608842, −1.781387448962806, −1.261018907688362, −0.4391615337022185, 0.4391615337022185, 1.261018907688362, 1.781387448962806, 2.422841196608842, 3.213833168171274, 3.769911024965562, 4.015197241347150, 4.793316890450383, 5.482989704200729, 5.731102940099073, 6.480596833828254, 6.714204986471061, 7.552899562667763, 7.823308242891913, 8.390930751673256, 8.982002846163236, 9.244948603106116, 10.02758741544144, 10.28263640533346, 10.71616089464343, 11.42403267706662, 11.88064543471111, 12.11152099088996, 12.64660971198113, 13.22216015109653