# Properties

 Label 2-546-13.10-c1-0-2 Degree $2$ Conductor $546$ Sign $-0.265 - 0.964i$ 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 + (−0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s − 0.732i·5-s + (−0.866 − 0.499i)6-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (0.366 + 0.633i)10-s + (−3.23 + 1.86i)11-s + 0.999·12-s + (−0.866 + 3.5i)13-s − 0.999·14-s + (0.633 − 0.366i)15-s + (−0.5 − 0.866i)16-s + (0.133 − 0.232i)17-s + ⋯
 L(s)  = 1 + (−0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s − 0.327i·5-s + (−0.353 − 0.204i)6-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.115 + 0.200i)10-s + (−0.974 + 0.562i)11-s + 0.288·12-s + (−0.240 + 0.970i)13-s − 0.267·14-s + (0.163 − 0.0945i)15-s + (−0.125 − 0.216i)16-s + (0.0324 − 0.0562i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.265 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$546$$    =    $$2 \cdot 3 \cdot 7 \cdot 13$$ Sign: $-0.265 - 0.964i$ Analytic conductor: $$4.35983$$ Root analytic conductor: $$2.08802$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{546} (127, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 546,\ (\ :1/2),\ -0.265 - 0.964i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.635970 + 0.834337i$$ $$L(\frac12)$$ $$\approx$$ $$0.635970 + 0.834337i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.866 - 0.5i)T$$
3 $$1 + (-0.5 - 0.866i)T$$
7 $$1 + (-0.866 - 0.5i)T$$
13 $$1 + (0.866 - 3.5i)T$$
good5 $$1 + 0.732iT - 5T^{2}$$
11 $$1 + (3.23 - 1.86i)T + (5.5 - 9.52i)T^{2}$$
17 $$1 + (-0.133 + 0.232i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-3.86 - 2.23i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 7.66iT - 31T^{2}$$
37 $$1 + (-1.09 + 0.633i)T + (18.5 - 32.0i)T^{2}$$
41 $$1 + (6.06 - 3.5i)T + (20.5 - 35.5i)T^{2}$$
43 $$1 + (-0.366 + 0.633i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 4.46iT - 47T^{2}$$
53 $$1 + 10.4T + 53T^{2}$$
59 $$1 + (-0.803 - 0.464i)T + (29.5 + 51.0i)T^{2}$$
61 $$1 + (-5.86 + 10.1i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (11.1 - 6.46i)T + (33.5 - 58.0i)T^{2}$$
71 $$1 + (-1.90 - 1.09i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + 6.53iT - 73T^{2}$$
79 $$1 - 10.8T + 79T^{2}$$
83 $$1 - 5.66iT - 83T^{2}$$
89 $$1 + (-5.59 + 3.23i)T + (44.5 - 77.0i)T^{2}$$
97 $$1 + (-0.633 - 0.366i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$