# Properties

 Label 2-273-13.3-c1-0-1 Degree $2$ Conductor $273$ Sign $-1$ Analytic cond. $2.17991$ Root an. cond. $1.47645$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (1.18 + 2.05i)2-s + (0.5 + 0.866i)3-s + (−1.82 + 3.16i)4-s − 4.02·5-s + (−1.18 + 2.05i)6-s + (0.5 − 0.866i)7-s − 3.92·8-s + (−0.499 + 0.866i)9-s + (−4.78 − 8.29i)10-s + (1.63 + 2.83i)11-s − 3.65·12-s + (0.910 + 3.48i)13-s + 2.37·14-s + (−2.01 − 3.48i)15-s + (−1.01 − 1.75i)16-s + (0.188 − 0.326i)17-s + ⋯
 L(s)  = 1 + (0.840 + 1.45i)2-s + (0.288 + 0.499i)3-s + (−0.912 + 1.58i)4-s − 1.80·5-s + (−0.485 + 0.840i)6-s + (0.188 − 0.327i)7-s − 1.38·8-s + (−0.166 + 0.288i)9-s + (−1.51 − 2.62i)10-s + (0.493 + 0.854i)11-s − 1.05·12-s + (0.252 + 0.967i)13-s + 0.635·14-s + (−0.520 − 0.900i)15-s + (−0.253 − 0.439i)16-s + (0.0457 − 0.0792i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$273$$    =    $$3 \cdot 7 \cdot 13$$ Sign: $-1$ Analytic conductor: $$2.17991$$ Root analytic conductor: $$1.47645$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{273} (211, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 273,\ (\ :1/2),\ -1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$-1.52306i$$ $$L(\frac12)$$ $$\approx$$ $$-1.52306i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.5 - 0.866i)T$$
7 $$1 + (-0.5 + 0.866i)T$$
13 $$1 + (-0.910 - 3.48i)T$$
good2 $$1 + (-1.18 - 2.05i)T + (-1 + 1.73i)T^{2}$$
5 $$1 + 4.02T + 5T^{2}$$
11 $$1 + (-1.63 - 2.83i)T + (-5.5 + 9.52i)T^{2}$$
17 $$1 + (-0.188 + 0.326i)T + (-8.5 - 14.7i)T^{2}$$
19 $$1 + (-1.77 + 3.07i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (0.948 + 1.64i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (-4.33 - 7.51i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 - 6.33T + 31T^{2}$$
37 $$1 + (2.01 + 3.48i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (3.75 + 6.50i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (-4.32 + 7.49i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + 12.1T + 47T^{2}$$
53 $$1 - 7.75T + 53T^{2}$$
59 $$1 + (-1.05 + 1.82i)T + (-29.5 - 51.0i)T^{2}$$
61 $$1 + (3.87 - 6.71i)T + (-30.5 - 52.8i)T^{2}$$
67 $$1 + (2.79 + 4.83i)T + (-33.5 + 58.0i)T^{2}$$
71 $$1 + (1.99 - 3.45i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 - 7.50T + 73T^{2}$$
79 $$1 + 1.16T + 79T^{2}$$
83 $$1 - 15.1T + 83T^{2}$$
89 $$1 + (3.17 + 5.50i)T + (-44.5 + 77.0i)T^{2}$$
97 $$1 + (2.48 - 4.30i)T + (-48.5 - 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$