# Properties

 Label 2-507-13.12-c1-0-17 Degree $2$ Conductor $507$ Sign $0.246 + 0.969i$ Analytic cond. $4.04841$ Root an. cond. $2.01206$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.24i·2-s + 3-s + 0.445·4-s + 2.80i·5-s − 1.24i·6-s − 4.80i·7-s − 3.04i·8-s + 9-s + 3.49·10-s + 1.46i·11-s + 0.445·12-s − 5.98·14-s + 2.80i·15-s − 2.91·16-s + 2.44·17-s − 1.24i·18-s + ⋯
 L(s)  = 1 − 0.881i·2-s + 0.577·3-s + 0.222·4-s + 1.25i·5-s − 0.509i·6-s − 1.81i·7-s − 1.07i·8-s + 0.333·9-s + 1.10·10-s + 0.442i·11-s + 0.128·12-s − 1.60·14-s + 0.723i·15-s − 0.727·16-s + 0.593·17-s − 0.293i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $0.246 + 0.969i$ Analytic conductor: $$4.04841$$ Root analytic conductor: $$2.01206$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{507} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 507,\ (\ :1/2),\ 0.246 + 0.969i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.57274 - 1.22231i$$ $$L(\frac12)$$ $$\approx$$ $$1.57274 - 1.22231i$$ $$L(\frac{3}{2})$$ 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$$
13 $$1$$
good2 $$1 + 1.24iT - 2T^{2}$$
5 $$1 - 2.80iT - 5T^{2}$$
7 $$1 + 4.80iT - 7T^{2}$$
11 $$1 - 1.46iT - 11T^{2}$$
17 $$1 - 2.44T + 17T^{2}$$
19 $$1 + 2.54iT - 19T^{2}$$
23 $$1 - 3.51T + 23T^{2}$$
29 $$1 - 1.85T + 29T^{2}$$
31 $$1 - 7.63iT - 31T^{2}$$
37 $$1 + 4.55iT - 37T^{2}$$
41 $$1 + 1.24iT - 41T^{2}$$
43 $$1 + 2.38T + 43T^{2}$$
47 $$1 - 12.8iT - 47T^{2}$$
53 $$1 + 8.85T + 53T^{2}$$
59 $$1 - 2.17iT - 59T^{2}$$
61 $$1 + 7.82T + 61T^{2}$$
67 $$1 - 3.58iT - 67T^{2}$$
71 $$1 - 8.83iT - 71T^{2}$$
73 $$1 + 7.69iT - 73T^{2}$$
79 $$1 + 4.02T + 79T^{2}$$
83 $$1 - 0.652iT - 83T^{2}$$
89 $$1 - 6.29iT - 89T^{2}$$
97 $$1 - 10.0iT - 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$