# Properties

 Label 2-1323-63.16-c1-0-30 Degree $2$ Conductor $1323$ Sign $-0.618 + 0.785i$ Analytic cond. $10.5642$ Root an. cond. $3.25026$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.119 − 0.207i)2-s + (0.971 − 1.68i)4-s − 1.18·5-s − 0.942·8-s + (0.141 + 0.244i)10-s + 3.70·11-s + (−0.5 − 0.866i)13-s + (−1.83 − 3.16i)16-s + (−3.47 − 6.01i)17-s + (−0.971 + 1.68i)19-s + (−1.14 + 1.98i)20-s + (−0.442 − 0.766i)22-s + 5.60·23-s − 3.60·25-s + (−0.119 + 0.207i)26-s + ⋯
 L(s)  = 1 + (−0.0845 − 0.146i)2-s + (0.485 − 0.841i)4-s − 0.528·5-s − 0.333·8-s + (0.0446 + 0.0774i)10-s + 1.11·11-s + (−0.138 − 0.240i)13-s + (−0.457 − 0.792i)16-s + (−0.841 − 1.45i)17-s + (−0.222 + 0.385i)19-s + (−0.256 + 0.444i)20-s + (−0.0944 − 0.163i)22-s + 1.16·23-s − 0.720·25-s + (−0.0234 + 0.0406i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.618 + 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$1323$$    =    $$3^{3} \cdot 7^{2}$$ Sign: $-0.618 + 0.785i$ Analytic conductor: $$10.5642$$ Root analytic conductor: $$3.25026$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{1323} (667, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1323,\ (\ :1/2),\ -0.618 + 0.785i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.278331976$$ $$L(\frac12)$$ $$\approx$$ $$1.278331976$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1$$
good2 $$1 + (0.119 + 0.207i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + 1.18T + 5T^{2}$$
11 $$1 - 3.70T + 11T^{2}$$
13 $$1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (3.47 + 6.01i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (0.971 - 1.68i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 - 5.60T + 23T^{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 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{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 + (5.80 + 10.0i)T + (-26.5 + 45.8i)T^{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 + (7.57 + 13.1i)T + (-36.5 + 63.2i)T^{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 + (-1.37 + 2.37i)T + (-44.5 - 77.0i)T^{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}$$