# Properties

 Label 2-240-3.2-c2-0-14 Degree $2$ Conductor $240$ Sign $-0.995 + 0.0972i$ Analytic cond. $6.53952$ Root an. cond. $2.55724$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.291 − 2.98i)3-s + 2.23i·5-s − 4.46·7-s + (−8.82 + 1.74i)9-s − 17.8i·11-s − 11.0·13-s + (6.67 − 0.652i)15-s − 0.794i·17-s − 26.5·19-s + (1.30 + 13.3i)21-s + 14.9i·23-s − 5.00·25-s + (7.77 + 25.8i)27-s − 5.58i·29-s − 53.1·31-s + ⋯
 L(s)  = 1 + (−0.0972 − 0.995i)3-s + 0.447i·5-s − 0.637·7-s + (−0.981 + 0.193i)9-s − 1.62i·11-s − 0.846·13-s + (0.445 − 0.0434i)15-s − 0.0467i·17-s − 1.39·19-s + (0.0619 + 0.634i)21-s + 0.649i·23-s − 0.200·25-s + (0.287 + 0.957i)27-s − 0.192i·29-s − 1.71·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$240$$    =    $$2^{4} \cdot 3 \cdot 5$$ Sign: $-0.995 + 0.0972i$ Analytic conductor: $$6.53952$$ Root analytic conductor: $$2.55724$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{240} (161, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 240,\ (\ :1),\ -0.995 + 0.0972i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.0269264 - 0.552710i$$ $$L(\frac12)$$ $$\approx$$ $$0.0269264 - 0.552710i$$ $$L(2)$$ 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 + (0.291 + 2.98i)T$$
5 $$1 - 2.23iT$$
good7 $$1 + 4.46T + 49T^{2}$$
11 $$1 + 17.8iT - 121T^{2}$$
13 $$1 + 11.0T + 169T^{2}$$
17 $$1 + 0.794iT - 289T^{2}$$
19 $$1 + 26.5T + 361T^{2}$$
23 $$1 - 14.9iT - 529T^{2}$$
29 $$1 + 5.58iT - 841T^{2}$$
31 $$1 + 53.1T + 961T^{2}$$
37 $$1 - 51.7T + 1.36e3T^{2}$$
41 $$1 + 67.8iT - 1.68e3T^{2}$$
43 $$1 - 40.8T + 1.84e3T^{2}$$
47 $$1 - 12.3iT - 2.20e3T^{2}$$
53 $$1 + 37.0iT - 2.80e3T^{2}$$
59 $$1 + 61.0iT - 3.48e3T^{2}$$
61 $$1 - 97.8T + 3.72e3T^{2}$$
67 $$1 + 3.02T + 4.48e3T^{2}$$
71 $$1 + 57.0iT - 5.04e3T^{2}$$
73 $$1 + 31.4T + 5.32e3T^{2}$$
79 $$1 - 2.16T + 6.24e3T^{2}$$
83 $$1 + 13.0iT - 6.88e3T^{2}$$
89 $$1 - 173. iT - 7.92e3T^{2}$$
97 $$1 + 91.6T + 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$