# Properties

 Label 2-546-273.269-c1-0-4 Degree $2$ Conductor $546$ Sign $0.433 - 0.901i$ 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 − i·2-s + (−1.42 − 0.988i)3-s − 4-s + (1.41 + 2.44i)5-s + (−0.988 + 1.42i)6-s + (0.383 − 2.61i)7-s + i·8-s + (1.04 + 2.81i)9-s + (2.44 − 1.41i)10-s + (−4.64 + 2.68i)11-s + (1.42 + 0.988i)12-s + (−2.04 + 2.97i)13-s + (−2.61 − 0.383i)14-s + (0.408 − 4.88i)15-s + 16-s − 7.32·17-s + ⋯
 L(s)  = 1 − 0.707i·2-s + (−0.821 − 0.570i)3-s − 0.5·4-s + (0.632 + 1.09i)5-s + (−0.403 + 0.580i)6-s + (0.145 − 0.989i)7-s + 0.353i·8-s + (0.349 + 0.937i)9-s + (0.774 − 0.447i)10-s + (−1.40 + 0.808i)11-s + (0.410 + 0.285i)12-s + (−0.565 + 0.824i)13-s + (−0.699 − 0.102i)14-s + (0.105 − 1.26i)15-s + 0.250·16-s − 1.77·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.433 - 0.901i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (269, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.433 - 0.901i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.495241 + 0.311291i$$ $$L(\frac12)$$ $$\approx$$ $$0.495241 + 0.311291i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + (1.42 + 0.988i)T$$
7 $$1 + (-0.383 + 2.61i)T$$
13 $$1 + (2.04 - 2.97i)T$$
good5 $$1 + (-1.41 - 2.44i)T + (-2.5 + 4.33i)T^{2}$$
11 $$1 + (4.64 - 2.68i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + 7.32T + 17T^{2}$$
19 $$1 + (-2.17 - 1.25i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 - 4.84iT - 23T^{2}$$
29 $$1 + (-5.21 - 3.01i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (-1.76 - 1.01i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 - 0.639T + 37T^{2}$$
41 $$1 + (5.97 - 10.3i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 + (0.836 + 1.44i)T + (-21.5 + 37.2i)T^{2}$$
47 $$1 + (2.48 + 4.31i)T + (-23.5 + 40.7i)T^{2}$$
53 $$1 + (-2.82 - 1.63i)T + (26.5 + 45.8i)T^{2}$$
59 $$1 - 4.67T + 59T^{2}$$
61 $$1 + (-4.84 - 2.79i)T + (30.5 + 52.8i)T^{2}$$
67 $$1 + (2.19 + 3.79i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (-7.14 + 4.12i)T + (35.5 - 61.4i)T^{2}$$
73 $$1 + (-1.55 - 0.896i)T + (36.5 + 63.2i)T^{2}$$
79 $$1 + (-2.02 - 3.51i)T + (-39.5 + 68.4i)T^{2}$$
83 $$1 + 12.1T + 83T^{2}$$
89 $$1 + 16.2T + 89T^{2}$$
97 $$1 + (4.57 - 2.64i)T + (48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$