# Properties

 Label 2-546-13.12-c1-0-8 Degree $2$ Conductor $546$ Sign $0.832 - 0.554i$ 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 + 3-s − 4-s − i·5-s + i·6-s − i·7-s − i·8-s + 9-s + 10-s + 3i·11-s − 12-s + (3 − 2i)13-s + 14-s − i·15-s + 16-s + 7·17-s + ⋯
 L(s)  = 1 + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s − 0.377i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 0.904i·11-s − 0.288·12-s + (0.832 − 0.554i)13-s + 0.267·14-s − 0.258i·15-s + 0.250·16-s + 1.69·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $0.832 - 0.554i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ 0.832 - 0.554i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.70292 + 0.515605i$$ $$L(\frac12)$$ $$\approx$$ $$1.70292 + 0.515605i$$ $$L(\frac{3}{2})$$ 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 - T$$
7 $$1 + iT$$
13 $$1 + (-3 + 2i)T$$
good5 $$1 + iT - 5T^{2}$$
11 $$1 - 3iT - 11T^{2}$$
17 $$1 - 7T + 17T^{2}$$
19 $$1 - 3iT - 19T^{2}$$
23 $$1 - T + 23T^{2}$$
29 $$1 + T + 29T^{2}$$
31 $$1 + 8iT - 31T^{2}$$
37 $$1 + iT - 37T^{2}$$
41 $$1 - 4iT - 41T^{2}$$
43 $$1 + 5T + 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 + 6T + 53T^{2}$$
59 $$1 - 10iT - 59T^{2}$$
61 $$1 + 13T + 61T^{2}$$
67 $$1 + 8iT - 67T^{2}$$
71 $$1 - 6iT - 71T^{2}$$
73 $$1 + 13iT - 73T^{2}$$
79 $$1 + 12T + 79T^{2}$$
83 $$1 - 2iT - 83T^{2}$$
89 $$1 - 12iT - 89T^{2}$$
97 $$1 - 6iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$