# Properties

 Degree $2$ Conductor $63$ Sign $0.787 - 0.616i$ Motivic weight $1$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.119 − 0.207i)2-s + (−0.619 + 1.61i)3-s + (0.971 + 1.68i)4-s + (−0.590 − 1.02i)5-s + (0.260 + 0.321i)6-s + (0.5 − 0.866i)7-s + 0.942·8-s + (−2.23 − 2.00i)9-s − 0.282·10-s + (1.85 − 3.20i)11-s + (−3.32 + 0.528i)12-s + (−0.5 − 0.866i)13-s + (−0.119 − 0.207i)14-s + (2.02 − 0.321i)15-s + (−1.83 + 3.16i)16-s − 6.94·17-s + ⋯
 L(s)  = 1 + (0.0845 − 0.146i)2-s + (−0.357 + 0.933i)3-s + (0.485 + 0.841i)4-s + (−0.264 − 0.457i)5-s + (0.106 + 0.131i)6-s + (0.188 − 0.327i)7-s + 0.333·8-s + (−0.744 − 0.668i)9-s − 0.0893·10-s + (0.558 − 0.967i)11-s + (−0.959 + 0.152i)12-s + (−0.138 − 0.240i)13-s + (−0.0319 − 0.0553i)14-s + (0.522 − 0.0830i)15-s + (−0.457 + 0.792i)16-s − 1.68·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.787 - 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$63$$    =    $$3^{2} \cdot 7$$ Sign: $0.787 - 0.616i$ Motivic weight: $$1$$ Character: $\chi_{63} (43, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 63,\ (\ :1/2),\ 0.787 - 0.616i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.839017 + 0.289564i$$ $$L(\frac12)$$ $$\approx$$ $$0.839017 + 0.289564i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (0.619 - 1.61i)T$$
7 $$1 + (-0.5 + 0.866i)T$$
good2 $$1 + (-0.119 + 0.207i)T + (-1 - 1.73i)T^{2}$$
5 $$1 + (0.590 + 1.02i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (-1.85 + 3.20i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + 6.94T + 17T^{2}$$
19 $$1 - 1.94T + 19T^{2}$$
23 $$1 + (-2.80 - 4.85i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (0.119 - 0.207i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (0.830 + 1.43i)T + (-15.5 + 26.8i)T^{2}$$
37 $$1 + 9.54T + 37T^{2}$$
41 $$1 + (-5.09 - 8.81i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (1.11 - 1.92i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (2.91 - 5.04i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + 11.6T + 53T^{2}$$
59 $$1 + (1.30 + 2.25i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + (-3.80 + 6.58i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (1.75 + 3.03i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 - 8.60T + 71T^{2}$$
73 $$1 - 15.1T + 73T^{2}$$
79 $$1 + (3.68 - 6.38i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-3.47 + 6.01i)T + (-41.5 - 71.8i)T^{2}$$
89 $$1 - 2.74T + 89T^{2}$$
97 $$1 + (3.58 - 6.20i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$