# Properties

 Label 2-8280-1.1-c1-0-81 Degree $2$ Conductor $8280$ Sign $-1$ Analytic cond. $66.1161$ Root an. cond. $8.13118$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $1$

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## Dirichlet series

 L(s)  = 1 − 5-s + 2·7-s − 4·11-s + 2·13-s + 2·17-s − 8·19-s − 23-s + 25-s + 4·29-s + 8·31-s − 2·35-s + 8·37-s + 4·41-s − 10·43-s − 8·47-s − 3·49-s − 6·53-s + 4·55-s − 10·59-s + 10·61-s − 2·65-s + 2·67-s − 2·71-s + 2·73-s − 8·77-s + 16·79-s + 12·83-s + ⋯
 L(s)  = 1 − 0.447·5-s + 0.755·7-s − 1.20·11-s + 0.554·13-s + 0.485·17-s − 1.83·19-s − 0.208·23-s + 1/5·25-s + 0.742·29-s + 1.43·31-s − 0.338·35-s + 1.31·37-s + 0.624·41-s − 1.52·43-s − 1.16·47-s − 3/7·49-s − 0.824·53-s + 0.539·55-s − 1.30·59-s + 1.28·61-s − 0.248·65-s + 0.244·67-s − 0.237·71-s + 0.234·73-s − 0.911·77-s + 1.80·79-s + 1.31·83-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$8280$$    =    $$2^{3} \cdot 3^{2} \cdot 5 \cdot 23$$ Sign: $-1$ Analytic conductor: $$66.1161$$ Root analytic conductor: $$8.13118$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$1$$ Selberg data: $$(2,\ 8280,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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 + T$$
23 $$1 + T$$
good7 $$1 - 2 T + p T^{2}$$
11 $$1 + 4 T + p T^{2}$$
13 $$1 - 2 T + p T^{2}$$
17 $$1 - 2 T + p T^{2}$$
19 $$1 + 8 T + p T^{2}$$
29 $$1 - 4 T + p T^{2}$$
31 $$1 - 8 T + p T^{2}$$
37 $$1 - 8 T + p T^{2}$$
41 $$1 - 4 T + p T^{2}$$
43 $$1 + 10 T + p T^{2}$$
47 $$1 + 8 T + p T^{2}$$
53 $$1 + 6 T + p T^{2}$$
59 $$1 + 10 T + p T^{2}$$
61 $$1 - 10 T + p T^{2}$$
67 $$1 - 2 T + p T^{2}$$
71 $$1 + 2 T + p T^{2}$$
73 $$1 - 2 T + p T^{2}$$
79 $$1 - 16 T + p T^{2}$$
83 $$1 - 12 T + p T^{2}$$
89 $$1 + 10 T + p T^{2}$$
97 $$1 + 8 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

−7.76986196859910930189726353998, −6.61379405843925092220884012653, −6.25337390344217045262238524439, −5.19701860404870808207037752942, −4.68569198529467664866740337253, −4.01058367021528819304449214602, −3.01049763302551176117126573989, −2.28335328675673742864312113123, −1.23911131035806829740524494216, 0, 1.23911131035806829740524494216, 2.28335328675673742864312113123, 3.01049763302551176117126573989, 4.01058367021528819304449214602, 4.68569198529467664866740337253, 5.19701860404870808207037752942, 6.25337390344217045262238524439, 6.61379405843925092220884012653, 7.76986196859910930189726353998