# Properties

 Label 2-30-15.14-c2-0-2 Degree $2$ Conductor $30$ Sign $0.668 + 0.744i$ Analytic cond. $0.817440$ Root an. cond. $0.904124$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41·2-s + (0.707 − 2.91i)3-s + 2.00·4-s + (2.82 − 4.12i)5-s + (−1.00 + 4.12i)6-s + 5.83i·7-s − 2.82·8-s + (−8 − 4.12i)9-s + (−4.00 + 5.83i)10-s + 16.4i·11-s + (1.41 − 5.83i)12-s − 8.24i·14-s + (−10.0 − 11.1i)15-s + 4.00·16-s + 11.3·17-s + (11.3 + 5.83i)18-s + ⋯
 L(s)  = 1 − 0.707·2-s + (0.235 − 0.971i)3-s + 0.500·4-s + (0.565 − 0.824i)5-s + (−0.166 + 0.687i)6-s + 0.832i·7-s − 0.353·8-s + (−0.888 − 0.458i)9-s + (−0.400 + 0.583i)10-s + 1.49i·11-s + (0.117 − 0.485i)12-s − 0.589i·14-s + (−0.668 − 0.744i)15-s + 0.250·16-s + 0.665·17-s + (0.628 + 0.323i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.668 + 0.744i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 30 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.668 + 0.744i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$30$$    =    $$2 \cdot 3 \cdot 5$$ Sign: $0.668 + 0.744i$ Analytic conductor: $$0.817440$$ Root analytic conductor: $$0.904124$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{30} (29, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 30,\ (\ :1),\ 0.668 + 0.744i)$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.751356 - 0.335177i$$ $$L(\frac12)$$ $$\approx$$ $$0.751356 - 0.335177i$$ $$L(2)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 1.41T$$
3 $$1 + (-0.707 + 2.91i)T$$
5 $$1 + (-2.82 + 4.12i)T$$
good7 $$1 - 5.83iT - 49T^{2}$$
11 $$1 - 16.4iT - 121T^{2}$$
13 $$1 - 169T^{2}$$
17 $$1 - 11.3T + 289T^{2}$$
19 $$1 - 12T + 361T^{2}$$
23 $$1 + 24.0T + 529T^{2}$$
29 $$1 - 841T^{2}$$
31 $$1 + 32T + 961T^{2}$$
37 $$1 - 23.3iT - 1.36e3T^{2}$$
41 $$1 + 57.7iT - 1.68e3T^{2}$$
43 $$1 + 40.8iT - 1.84e3T^{2}$$
47 $$1 - 35.3T + 2.20e3T^{2}$$
53 $$1 + 67.8T + 2.80e3T^{2}$$
59 $$1 - 16.4iT - 3.48e3T^{2}$$
61 $$1 + 16T + 3.72e3T^{2}$$
67 $$1 + 5.83iT - 4.48e3T^{2}$$
71 $$1 - 5.04e3T^{2}$$
73 $$1 - 116. iT - 5.32e3T^{2}$$
79 $$1 + 72T + 6.24e3T^{2}$$
83 $$1 - 43.8T + 6.88e3T^{2}$$
89 $$1 + 65.9iT - 7.92e3T^{2}$$
97 $$1 + 163. iT - 9.40e3T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$