# Properties

 Label 2-18e2-36.23-c1-0-7 Degree $2$ Conductor $324$ Sign $0.995 + 0.0962i$ Analytic cond. $2.58715$ Root an. cond. $1.60846$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.40 − 0.190i)2-s + (1.92 + 0.535i)4-s + (1.93 + 1.11i)5-s + (3.35 − 1.93i)7-s + (−2.59 − 1.11i)8-s + (−2.5 − 1.93i)10-s + (−0.866 − 1.5i)11-s + (−1 + 1.73i)13-s + (−5.06 + 2.07i)14-s + (3.42 + 2.06i)16-s + 4.47i·17-s + (3.13 + 3.19i)20-s + (0.927 + 2.26i)22-s + (3.46 − 6i)23-s + (1.73 − 2.23i)26-s + ⋯
 L(s)  = 1 + (−0.990 − 0.135i)2-s + (0.963 + 0.267i)4-s + (0.866 + 0.499i)5-s + (1.26 − 0.731i)7-s + (−0.918 − 0.395i)8-s + (−0.790 − 0.612i)10-s + (−0.261 − 0.452i)11-s + (−0.277 + 0.480i)13-s + (−1.35 + 0.554i)14-s + (0.856 + 0.515i)16-s + 1.08i·17-s + (0.700 + 0.713i)20-s + (0.197 + 0.483i)22-s + (0.722 − 1.25i)23-s + (0.339 − 0.438i)26-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0962i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0962i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$324$$    =    $$2^{2} \cdot 3^{4}$$ Sign: $0.995 + 0.0962i$ Analytic conductor: $$2.58715$$ Root analytic conductor: $$1.60846$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{324} (215, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 324,\ (\ :1/2),\ 0.995 + 0.0962i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.10575 - 0.0533308i$$ $$L(\frac12)$$ $$\approx$$ $$1.10575 - 0.0533308i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (1.40 + 0.190i)T$$
3 $$1$$
good5 $$1 + (-1.93 - 1.11i)T + (2.5 + 4.33i)T^{2}$$
7 $$1 + (-3.35 + 1.93i)T + (3.5 - 6.06i)T^{2}$$
11 $$1 + (0.866 + 1.5i)T + (-5.5 + 9.52i)T^{2}$$
13 $$1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2}$$
17 $$1 - 4.47iT - 17T^{2}$$
19 $$1 - 19T^{2}$$
23 $$1 + (-3.46 + 6i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (-3.87 + 2.23i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (-3.35 - 1.93i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + 4T + 37T^{2}$$
41 $$1 + (-7.74 - 4.47i)T + (20.5 + 35.5i)T^{2}$$
43 $$1 + (6.70 - 3.87i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (-1.73 - 3i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + 2.23iT - 53T^{2}$$
59 $$1 + (1.73 - 3i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (6.70 + 3.87i)T + (33.5 + 58.0i)T^{2}$$
71 $$1 + 10.3T + 71T^{2}$$
73 $$1 - 5T + 73T^{2}$$
79 $$1 + (6.70 - 3.87i)T + (39.5 - 68.4i)T^{2}$$
83 $$1 + (6.06 + 10.5i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 - 4.47iT - 89T^{2}$$
97 $$1 + (5.5 + 9.52i)T + (-48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$