# Properties

 Label 2-546-273.272-c1-0-6 Degree $2$ Conductor $546$ Sign $-0.744 - 0.667i$ 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 + (1.26 + 1.18i)3-s + 4-s + 3.37i·5-s + (−1.26 − 1.18i)6-s + (−2.52 + 0.792i)7-s − 8-s + (0.186 + 2.99i)9-s − 3.37i·10-s + 5.74·11-s + (1.26 + 1.18i)12-s + (−3.46 − i)13-s + (2.52 − 0.792i)14-s + (−4 + 4.25i)15-s + 16-s − 0.792·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (0.728 + 0.684i)3-s + 0.5·4-s + 1.50i·5-s + (−0.515 − 0.484i)6-s + (−0.954 + 0.299i)7-s − 0.353·8-s + (0.0620 + 0.998i)9-s − 1.06i·10-s + 1.73·11-s + (0.364 + 0.342i)12-s + (−0.960 − 0.277i)13-s + (0.674 − 0.211i)14-s + (−1.03 + 1.09i)15-s + 0.250·16-s − 0.192·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.744 - 0.667i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.744 - 0.667i$ 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: $$(2,\ 546,\ (\ :1/2),\ -0.744 - 0.667i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.395712 + 1.03360i$$ $$L(\frac12)$$ $$\approx$$ $$0.395712 + 1.03360i$$ $$L(\frac{3}{2})$$ 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 + (-1.26 - 1.18i)T$$
7 $$1 + (2.52 - 0.792i)T$$
13 $$1 + (3.46 + i)T$$
good5 $$1 - 3.37iT - 5T^{2}$$
11 $$1 - 5.74T + 11T^{2}$$
17 $$1 + 0.792T + 17T^{2}$$
19 $$1 + 2.37T + 19T^{2}$$
23 $$1 - 0.147iT - 23T^{2}$$
29 $$1 - 2.37iT - 29T^{2}$$
31 $$1 - 4.10T + 31T^{2}$$
37 $$1 + 8.36iT - 37T^{2}$$
41 $$1 + 8.37iT - 41T^{2}$$
43 $$1 + 2.62T + 43T^{2}$$
47 $$1 - 10.3iT - 47T^{2}$$
53 $$1 - 10.0iT - 53T^{2}$$
59 $$1 - 2.74iT - 59T^{2}$$
61 $$1 - 7.74iT - 61T^{2}$$
67 $$1 - 5.98iT - 67T^{2}$$
71 $$1 - 2T + 71T^{2}$$
73 $$1 - 10.2T + 73T^{2}$$
79 $$1 + 3.62T + 79T^{2}$$
83 $$1 - 6.74iT - 83T^{2}$$
89 $$1 + 10iT - 89T^{2}$$
97 $$1 - 9.15T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$