# Properties

 Label 2-15-5.4-c3-0-0 Degree $2$ Conductor $15$ Sign $0.116 - 0.993i$ Analytic cond. $0.885028$ Root an. cond. $0.940759$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 4.70i·2-s − 3i·3-s − 14.1·4-s + (11.1 + 1.29i)5-s + 14.1·6-s − 16.2i·7-s − 28.7i·8-s − 9·9-s + (−6.10 + 52.2i)10-s − 40.2·11-s + 42.3i·12-s − 19.7i·13-s + 76.2·14-s + (3.89 − 33.3i)15-s + 22.1·16-s + 83.0i·17-s + ⋯
 L(s)  = 1 + 1.66i·2-s − 0.577i·3-s − 1.76·4-s + (0.993 + 0.116i)5-s + 0.959·6-s − 0.875i·7-s − 1.26i·8-s − 0.333·9-s + (−0.193 + 1.65i)10-s − 1.10·11-s + 1.01i·12-s − 0.422i·13-s + 1.45·14-s + (0.0670 − 0.573i)15-s + 0.345·16-s + 1.18i·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$15$$    =    $$3 \cdot 5$$ Sign: $0.116 - 0.993i$ Analytic conductor: $$0.885028$$ Root analytic conductor: $$0.940759$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{15} (4, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 15,\ (\ :3/2),\ 0.116 - 0.993i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.746621 + 0.664408i$$ $$L(\frac12)$$ $$\approx$$ $$0.746621 + 0.664408i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + 3iT$$
5 $$1 + (-11.1 - 1.29i)T$$
good2 $$1 - 4.70iT - 8T^{2}$$
7 $$1 + 16.2iT - 343T^{2}$$
11 $$1 + 40.2T + 1.33e3T^{2}$$
13 $$1 + 19.7iT - 2.19e3T^{2}$$
17 $$1 - 83.0iT - 4.91e3T^{2}$$
19 $$1 - 48.8T + 6.85e3T^{2}$$
23 $$1 + 1.61iT - 1.21e4T^{2}$$
29 $$1 - 24.5T + 2.43e4T^{2}$$
31 $$1 + 12.4T + 2.97e4T^{2}$$
37 $$1 - 325. iT - 5.06e4T^{2}$$
41 $$1 + 242.T + 6.89e4T^{2}$$
43 $$1 + 367. iT - 7.95e4T^{2}$$
47 $$1 + 204. iT - 1.03e5T^{2}$$
53 $$1 - 61.5iT - 1.48e5T^{2}$$
59 $$1 - 112.T + 2.05e5T^{2}$$
61 $$1 - 477.T + 2.26e5T^{2}$$
67 $$1 - 558. iT - 3.00e5T^{2}$$
71 $$1 - 558.T + 3.57e5T^{2}$$
73 $$1 + 1.01e3iT - 3.89e5T^{2}$$
79 $$1 + 1.15e3T + 4.93e5T^{2}$$
83 $$1 - 1.15e3iT - 5.71e5T^{2}$$
89 $$1 + 96.9T + 7.04e5T^{2}$$
97 $$1 + 1.15e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$