# Properties

 Label 2-273-273.68-c1-0-7 Degree $2$ Conductor $273$ Sign $-0.923 + 0.383i$ 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 + 2.39i·2-s + (1.36 + 1.05i)3-s − 3.74·4-s + (−1.10 + 1.90i)5-s + (−2.54 + 3.28i)6-s + (−1.62 − 2.08i)7-s − 4.18i·8-s + (0.753 + 2.90i)9-s + (−4.56 − 2.63i)10-s + (−1.30 − 0.751i)11-s + (−5.12 − 3.96i)12-s + (3.47 + 0.976i)13-s + (5.00 − 3.89i)14-s + (−3.52 + 1.44i)15-s + 2.53·16-s + 4.90·17-s + ⋯
 L(s)  = 1 + 1.69i·2-s + (0.790 + 0.611i)3-s − 1.87·4-s + (−0.492 + 0.852i)5-s + (−1.03 + 1.34i)6-s + (−0.614 − 0.789i)7-s − 1.47i·8-s + (0.251 + 0.967i)9-s + (−1.44 − 0.834i)10-s + (−0.392 − 0.226i)11-s + (−1.48 − 1.14i)12-s + (0.962 + 0.270i)13-s + (1.33 − 1.04i)14-s + (−0.911 + 0.373i)15-s + 0.632·16-s + 1.19·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$273$$    =    $$3 \cdot 7 \cdot 13$$ Sign: $-0.923 + 0.383i$ Analytic conductor: $$2.17991$$ Root analytic conductor: $$1.47645$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{273} (68, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 273,\ (\ :1/2),\ -0.923 + 0.383i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.243600 - 1.22262i$$ $$L(\frac12)$$ $$\approx$$ $$0.243600 - 1.22262i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-1.36 - 1.05i)T$$
7 $$1 + (1.62 + 2.08i)T$$
13 $$1 + (-3.47 - 0.976i)T$$
good2 $$1 - 2.39iT - 2T^{2}$$
5 $$1 + (1.10 - 1.90i)T + (-2.5 - 4.33i)T^{2}$$
11 $$1 + (1.30 + 0.751i)T + (5.5 + 9.52i)T^{2}$$
17 $$1 - 4.90T + 17T^{2}$$
19 $$1 + (2.77 - 1.59i)T + (9.5 - 16.4i)T^{2}$$
23 $$1 - 4.94iT - 23T^{2}$$
29 $$1 + (-0.775 + 0.447i)T + (14.5 - 25.1i)T^{2}$$
31 $$1 + (-2.33 + 1.34i)T + (15.5 - 26.8i)T^{2}$$
37 $$1 - 10.0T + 37T^{2}$$
41 $$1 + (4.98 + 8.63i)T + (-20.5 + 35.5i)T^{2}$$
43 $$1 + (0.820 - 1.42i)T + (-21.5 - 37.2i)T^{2}$$
47 $$1 + (-0.165 + 0.286i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 + (-5.36 + 3.09i)T + (26.5 - 45.8i)T^{2}$$
59 $$1 + 13.8T + 59T^{2}$$
61 $$1 + (-2.44 + 1.41i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-3.46 + 6.00i)T + (-33.5 - 58.0i)T^{2}$$
71 $$1 + (-13.9 - 8.04i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + (2.87 - 1.65i)T + (36.5 - 63.2i)T^{2}$$
79 $$1 + (3.02 - 5.23i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + 2.74T + 83T^{2}$$
89 $$1 - 12.6T + 89T^{2}$$
97 $$1 + (11.5 + 6.67i)T + (48.5 + 84.0i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$