# Properties

 Label 2-273-13.12-c1-0-2 Degree $2$ Conductor $273$ Sign $-0.0805 - 0.996i$ 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 + 1.29i·2-s − 3-s + 0.334·4-s − 1.33i·5-s − 1.29i·6-s + i·7-s + 3.01i·8-s + 9-s + 1.72·10-s + 4.88i·11-s − 0.334·12-s + (3.59 − 0.290i)13-s − 1.29·14-s + 1.33i·15-s − 3.21·16-s + 4.58·17-s + ⋯
 L(s)  = 1 + 0.912i·2-s − 0.577·3-s + 0.167·4-s − 0.596i·5-s − 0.526i·6-s + 0.377i·7-s + 1.06i·8-s + 0.333·9-s + 0.544·10-s + 1.47i·11-s − 0.0965·12-s + (0.996 − 0.0805i)13-s − 0.344·14-s + 0.344i·15-s − 0.804·16-s + 1.11·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0805 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$273$$    =    $$3 \cdot 7 \cdot 13$$ Sign: $-0.0805 - 0.996i$ Analytic conductor: $$2.17991$$ Root analytic conductor: $$1.47645$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{273} (64, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 273,\ (\ :1/2),\ -0.0805 - 0.996i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.829207 + 0.898943i$$ $$L(\frac12)$$ $$\approx$$ $$0.829207 + 0.898943i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + T$$
7 $$1 - iT$$
13 $$1 + (-3.59 + 0.290i)T$$
good2 $$1 - 1.29iT - 2T^{2}$$
5 $$1 + 1.33iT - 5T^{2}$$
11 $$1 - 4.88iT - 11T^{2}$$
17 $$1 - 4.58T + 17T^{2}$$
19 $$1 - 2.96iT - 19T^{2}$$
23 $$1 + 6.13T + 23T^{2}$$
29 $$1 + 7.63T + 29T^{2}$$
31 $$1 + 5.63iT - 31T^{2}$$
37 $$1 + 9.76iT - 37T^{2}$$
41 $$1 - 3.69iT - 41T^{2}$$
43 $$1 - 9.63T + 43T^{2}$$
47 $$1 + 5.27iT - 47T^{2}$$
53 $$1 - 10.7T + 53T^{2}$$
59 $$1 + 10.3iT - 59T^{2}$$
61 $$1 + 11.1T + 61T^{2}$$
67 $$1 - 5.50iT - 67T^{2}$$
71 $$1 + 4.88iT - 71T^{2}$$
73 $$1 - 4.45iT - 73T^{2}$$
79 $$1 + 3.54T + 79T^{2}$$
83 $$1 - 5.27iT - 83T^{2}$$
89 $$1 + 9.94iT - 89T^{2}$$
97 $$1 + 18.2iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$