# Properties

 Label 2-546-91.45-c1-0-17 Degree $2$ Conductor $546$ Sign $-0.567 + 0.823i$ 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 + (0.707 − 0.707i)2-s + (0.866 − 0.5i)3-s − 1.00i·4-s + (0.349 − 1.30i)5-s + (0.258 − 0.965i)6-s + (−2.23 − 1.41i)7-s + (−0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.676 − 1.17i)10-s + (0.147 − 0.549i)11-s + (−0.500 − 0.866i)12-s + (−3.33 − 1.36i)13-s + (−2.58 + 0.579i)14-s + (−0.349 − 1.30i)15-s − 1.00·16-s + 0.975·17-s + ⋯
 L(s)  = 1 + (0.499 − 0.499i)2-s + (0.499 − 0.288i)3-s − 0.500i·4-s + (0.156 − 0.584i)5-s + (0.105 − 0.394i)6-s + (−0.844 − 0.535i)7-s + (−0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (−0.213 − 0.370i)10-s + (0.0444 − 0.165i)11-s + (−0.144 − 0.250i)12-s + (−0.925 − 0.379i)13-s + (−0.689 + 0.154i)14-s + (−0.0903 − 0.337i)15-s − 0.250·16-s + 0.236·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.567 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.567 + 0.823i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (409, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.567 + 0.823i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.895118 - 1.70479i$$ $$L(\frac12)$$ $$\approx$$ $$0.895118 - 1.70479i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (-0.707 + 0.707i)T$$
3 $$1 + (-0.866 + 0.5i)T$$
7 $$1 + (2.23 + 1.41i)T$$
13 $$1 + (3.33 + 1.36i)T$$
good5 $$1 + (-0.349 + 1.30i)T + (-4.33 - 2.5i)T^{2}$$
11 $$1 + (-0.147 + 0.549i)T + (-9.52 - 5.5i)T^{2}$$
17 $$1 - 0.975T + 17T^{2}$$
19 $$1 + (-3.14 + 0.841i)T + (16.4 - 9.5i)T^{2}$$
23 $$1 + 0.0810iT - 23T^{2}$$
29 $$1 + (0.139 - 0.242i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-2.48 + 0.665i)T + (26.8 - 15.5i)T^{2}$$
37 $$1 + (-1.14 - 1.14i)T + 37iT^{2}$$
41 $$1 + (-2.23 + 0.599i)T + (35.5 - 20.5i)T^{2}$$
43 $$1 + (-9.48 + 5.47i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (5.55 + 1.48i)T + (40.7 + 23.5i)T^{2}$$
53 $$1 + (4.54 - 7.86i)T + (-26.5 - 45.8i)T^{2}$$
59 $$1 + (-5.05 + 5.05i)T - 59iT^{2}$$
61 $$1 + (-4.95 - 2.86i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (-2.09 - 0.561i)T + (58.0 + 33.5i)T^{2}$$
71 $$1 + (-0.466 - 0.124i)T + (61.4 + 35.5i)T^{2}$$
73 $$1 + (0.753 + 2.81i)T + (-63.2 + 36.5i)T^{2}$$
79 $$1 + (-1.45 - 2.51i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + (2.22 + 2.22i)T + 83iT^{2}$$
89 $$1 + (7.48 - 7.48i)T - 89iT^{2}$$
97 $$1 + (-4.34 + 16.2i)T + (-84.0 - 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$