# Properties

 Label 2-483-23.22-c2-0-27 Degree $2$ Conductor $483$ Sign $-0.207 + 0.978i$ Analytic cond. $13.1607$ Root an. cond. $3.62778$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.65·2-s − 1.73·3-s − 1.27·4-s − 4.21i·5-s + 2.85·6-s − 2.64i·7-s + 8.70·8-s + 2.99·9-s + 6.95i·10-s + 4.54i·11-s + 2.20·12-s + 7.59·13-s + 4.36i·14-s + 7.30i·15-s − 9.27·16-s − 3.10i·17-s + ⋯
 L(s)  = 1 − 0.825·2-s − 0.577·3-s − 0.318·4-s − 0.842i·5-s + 0.476·6-s − 0.377i·7-s + 1.08·8-s + 0.333·9-s + 0.695i·10-s + 0.413i·11-s + 0.183·12-s + 0.584·13-s + 0.311i·14-s + 0.486i·15-s − 0.579·16-s − 0.182i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.207 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$483$$    =    $$3 \cdot 7 \cdot 23$$ Sign: $-0.207 + 0.978i$ Analytic conductor: $$13.1607$$ Root analytic conductor: $$3.62778$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{483} (22, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 483,\ (\ :1),\ -0.207 + 0.978i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.6414260592$$ $$L(\frac12)$$ $$\approx$$ $$0.6414260592$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 1.73T$$
7 $$1 + 2.64iT$$
23 $$1 + (-22.5 - 4.76i)T$$
good2 $$1 + 1.65T + 4T^{2}$$
5 $$1 + 4.21iT - 25T^{2}$$
11 $$1 - 4.54iT - 121T^{2}$$
13 $$1 - 7.59T + 169T^{2}$$
17 $$1 + 3.10iT - 289T^{2}$$
19 $$1 - 12.4iT - 361T^{2}$$
29 $$1 + 6.35T + 841T^{2}$$
31 $$1 + 30.1T + 961T^{2}$$
37 $$1 + 62.2iT - 1.36e3T^{2}$$
41 $$1 - 58.9T + 1.68e3T^{2}$$
43 $$1 + 46.9iT - 1.84e3T^{2}$$
47 $$1 - 7.95T + 2.20e3T^{2}$$
53 $$1 - 24.5iT - 2.80e3T^{2}$$
59 $$1 + 32.6T + 3.48e3T^{2}$$
61 $$1 + 81.8iT - 3.72e3T^{2}$$
67 $$1 + 100. iT - 4.48e3T^{2}$$
71 $$1 + 40.6T + 5.04e3T^{2}$$
73 $$1 + 57.2T + 5.32e3T^{2}$$
79 $$1 + 116. iT - 6.24e3T^{2}$$
83 $$1 - 38.4iT - 6.88e3T^{2}$$
89 $$1 - 21.3iT - 7.92e3T^{2}$$
97 $$1 + 68.3iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$