# Properties

 Label 2-9280-1.1-c1-0-66 Degree $2$ Conductor $9280$ Sign $1$ Analytic cond. $74.1011$ Root an. cond. $8.60820$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2·3-s − 5-s + 4.82·7-s + 9-s + 0.828·11-s + 2·13-s + 2·15-s + 2.82·17-s − 4.82·19-s − 9.65·21-s + 3.17·23-s + 25-s + 4·27-s − 29-s − 6.48·31-s − 1.65·33-s − 4.82·35-s + 8.48·37-s − 4·39-s − 6·41-s − 6·43-s − 45-s + 11.6·47-s + 16.3·49-s − 5.65·51-s + 3.65·53-s − 0.828·55-s + ⋯
 L(s)  = 1 − 1.15·3-s − 0.447·5-s + 1.82·7-s + 0.333·9-s + 0.249·11-s + 0.554·13-s + 0.516·15-s + 0.685·17-s − 1.10·19-s − 2.10·21-s + 0.661·23-s + 0.200·25-s + 0.769·27-s − 0.185·29-s − 1.16·31-s − 0.288·33-s − 0.816·35-s + 1.39·37-s − 0.640·39-s − 0.937·41-s − 0.914·43-s − 0.149·45-s + 1.70·47-s + 2.33·49-s − 0.792·51-s + 0.502·53-s − 0.111·55-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$9280$$    =    $$2^{6} \cdot 5 \cdot 29$$ Sign: $1$ Analytic conductor: $$74.1011$$ Root analytic conductor: $$8.60820$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: Trivial Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 9280,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.592917156$$ $$L(\frac12)$$ $$\approx$$ $$1.592917156$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + T$$
29 $$1 + T$$
good3 $$1 + 2T + 3T^{2}$$
7 $$1 - 4.82T + 7T^{2}$$
11 $$1 - 0.828T + 11T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 - 2.82T + 17T^{2}$$
19 $$1 + 4.82T + 19T^{2}$$
23 $$1 - 3.17T + 23T^{2}$$
31 $$1 + 6.48T + 31T^{2}$$
37 $$1 - 8.48T + 37T^{2}$$
41 $$1 + 6T + 41T^{2}$$
43 $$1 + 6T + 43T^{2}$$
47 $$1 - 11.6T + 47T^{2}$$
53 $$1 - 3.65T + 53T^{2}$$
59 $$1 + 59T^{2}$$
61 $$1 - 3.65T + 61T^{2}$$
67 $$1 - 6.48T + 67T^{2}$$
71 $$1 - 15.3T + 71T^{2}$$
73 $$1 - 8.48T + 73T^{2}$$
79 $$1 - 2.48T + 79T^{2}$$
83 $$1 - 7.17T + 83T^{2}$$
89 $$1 + 7.65T + 89T^{2}$$
97 $$1 + 12.4T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$