# Properties

 Label 2-363-3.2-c2-0-32 Degree $2$ Conductor $363$ Sign $0.833 - 0.552i$ Analytic cond. $9.89103$ Root an. cond. $3.14500$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.5 − 1.65i)3-s + 4·4-s + 9.94i·5-s + (3.5 − 8.29i)9-s + (10 − 6.63i)12-s + (16.5 + 24.8i)15-s + 16·16-s + 39.7i·20-s + 29.8i·23-s − 74·25-s + (−4.99 − 26.5i)27-s + 37·31-s + (14 − 33.1i)36-s + 25·37-s + (82.4 + 34.8i)45-s + ⋯
 L(s)  = 1 + (0.833 − 0.552i)3-s + 4-s + 1.98i·5-s + (0.388 − 0.921i)9-s + (0.833 − 0.552i)12-s + (1.10 + 1.65i)15-s + 16-s + 1.98i·20-s + 1.29i·23-s − 2.95·25-s + (−0.185 − 0.982i)27-s + 1.19·31-s + (0.388 − 0.921i)36-s + 0.675·37-s + (1.83 + 0.773i)45-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 363 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.833 - 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$363$$    =    $$3 \cdot 11^{2}$$ Sign: $0.833 - 0.552i$ Analytic conductor: $$9.89103$$ Root analytic conductor: $$3.14500$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{363} (122, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 363,\ (\ :1),\ 0.833 - 0.552i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.65602 + 0.800820i$$ $$L(\frac12)$$ $$\approx$$ $$2.65602 + 0.800820i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-2.5 + 1.65i)T$$
11 $$1$$
good2 $$1 - 4T^{2}$$
5 $$1 - 9.94iT - 25T^{2}$$
7 $$1 + 49T^{2}$$
13 $$1 + 169T^{2}$$
17 $$1 - 289T^{2}$$
19 $$1 + 361T^{2}$$
23 $$1 - 29.8iT - 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 - 37T + 961T^{2}$$
37 $$1 - 25T + 1.36e3T^{2}$$
41 $$1 - 1.68e3T^{2}$$
43 $$1 + 1.84e3T^{2}$$
47 $$1 + 79.5iT - 2.20e3T^{2}$$
53 $$1 + 79.5iT - 2.80e3T^{2}$$
59 $$1 - 49.7iT - 3.48e3T^{2}$$
61 $$1 + 3.72e3T^{2}$$
67 $$1 + 35T + 4.48e3T^{2}$$
71 $$1 + 49.7iT - 5.04e3T^{2}$$
73 $$1 + 5.32e3T^{2}$$
79 $$1 + 6.24e3T^{2}$$
83 $$1 - 6.88e3T^{2}$$
89 $$1 + 149. iT - 7.92e3T^{2}$$
97 $$1 - 95T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$