# Properties

 Label 2-546-91.81-c1-0-15 Degree $2$ Conductor $546$ Sign $-0.994 - 0.107i$ Analytic cond. $4.35983$ Root an. cond. $2.08802$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.5 − 0.866i)2-s − 3-s + (−0.499 − 0.866i)4-s + (0.623 + 1.07i)5-s + (−0.5 + 0.866i)6-s + (−2.27 − 1.34i)7-s − 0.999·8-s + 9-s + 1.24·10-s − 2.49·11-s + (0.499 + 0.866i)12-s + (−0.785 − 3.51i)13-s + (−2.30 + 1.30i)14-s + (−0.623 − 1.07i)15-s + (−0.5 + 0.866i)16-s + (−0.247 − 0.428i)17-s + ⋯
 L(s)  = 1 + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (0.278 + 0.482i)5-s + (−0.204 + 0.353i)6-s + (−0.861 − 0.507i)7-s − 0.353·8-s + 0.333·9-s + 0.394·10-s − 0.752·11-s + (0.144 + 0.249i)12-s + (−0.217 − 0.976i)13-s + (−0.615 + 0.348i)14-s + (−0.160 − 0.278i)15-s + (−0.125 + 0.216i)16-s + (−0.0600 − 0.104i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.994 - 0.107i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (445, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.994 - 0.107i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0257968 + 0.476417i$$ $$L(\frac12)$$ $$\approx$$ $$0.0257968 + 0.476417i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.5 + 0.866i)T$$
3 $$1 + T$$
7 $$1 + (2.27 + 1.34i)T$$
13 $$1 + (0.785 + 3.51i)T$$
good5 $$1 + (-0.623 - 1.07i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + 2.49T + 11T^{2}$$
17 $$1 + (0.247 + 0.428i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + 7.67T + 19T^{2}$$
23 $$1 + (0.224 - 0.388i)T + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (3.71 + 6.42i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (4.06 - 7.03i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1.37 - 2.37i)T + (-18.5 - 32.0i)T^{2}$$
41 $$1 + (1.87 + 3.24i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-1.47 + 2.55i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-3.29 - 5.71i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-3.86 + 6.69i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-0.727 - 1.26i)T + (-29.5 + 51.0i)T^{2}$$
61 $$1 + 4.19T + 61T^{2}$$
67 $$1 - 0.0276T + 67T^{2}$$
71 $$1 + (-4.68 + 8.12i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (5.07 - 8.78i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (4.93 + 8.54i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 - 7.42T + 83T^{2}$$
89 $$1 + (-5.74 + 9.94i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (0.509 - 0.883i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$