# Properties

 Label 2-546-273.272-c1-0-35 Degree $2$ Conductor $546$ Sign $-0.0179 + 0.999i$ 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 + 2-s + (0.420 − 1.68i)3-s + 4-s − 3.36i·5-s + (0.420 − 1.68i)6-s + (2.37 + 1.16i)7-s + 8-s + (−2.64 − 1.41i)9-s − 3.36i·10-s + (0.420 − 1.68i)12-s + (−2.79 + 2.27i)13-s + (2.37 + 1.16i)14-s + (−5.64 − 1.41i)15-s + 16-s + 7.82·17-s + (−2.64 − 1.41i)18-s + ⋯
 L(s)  = 1 + 0.707·2-s + (0.242 − 0.970i)3-s + 0.5·4-s − 1.50i·5-s + (0.171 − 0.685i)6-s + (0.898 + 0.439i)7-s + 0.353·8-s + (−0.881 − 0.471i)9-s − 1.06i·10-s + (0.121 − 0.485i)12-s + (−0.775 + 0.631i)13-s + (0.635 + 0.311i)14-s + (−1.45 − 0.365i)15-s + 0.250·16-s + 1.89·17-s + (−0.623 − 0.333i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0179 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0179 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.0179 + 0.999i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (545, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.0179 + 0.999i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.72468 - 1.75591i$$ $$L(\frac12)$$ $$\approx$$ $$1.72468 - 1.75591i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 - T$$
3 $$1 + (-0.420 + 1.68i)T$$
7 $$1 + (-2.37 - 1.16i)T$$
13 $$1 + (2.79 - 2.27i)T$$
good5 $$1 + 3.36iT - 5T^{2}$$
11 $$1 + 11T^{2}$$
17 $$1 - 7.82T + 17T^{2}$$
19 $$1 + 5.59T + 19T^{2}$$
23 $$1 - 0.500iT - 23T^{2}$$
29 $$1 - 5.15iT - 29T^{2}$$
31 $$1 + 3.06T + 31T^{2}$$
37 $$1 - 2.32iT - 37T^{2}$$
41 $$1 + 9.87iT - 41T^{2}$$
43 $$1 - 8T + 43T^{2}$$
47 $$1 + 4.33iT - 47T^{2}$$
53 $$1 - 0.500iT - 53T^{2}$$
59 $$1 - 2.16iT - 59T^{2}$$
61 $$1 + 4.55iT - 61T^{2}$$
67 $$1 - 13.1iT - 67T^{2}$$
71 $$1 - 6.58T + 71T^{2}$$
73 $$1 - 12.5T + 73T^{2}$$
79 $$1 + 0.708T + 79T^{2}$$
83 $$1 - 11.2iT - 83T^{2}$$
89 $$1 - 3.14iT - 89T^{2}$$
97 $$1 + 10.8T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$