# Properties

 Label 2-507-13.12-c3-0-21 Degree $2$ Conductor $507$ Sign $-0.999 + 0.0304i$ Analytic cond. $29.9139$ Root an. cond. $5.46936$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 2.34i·2-s + 3·3-s + 2.49·4-s + 15.3i·5-s + 7.04i·6-s + 10.1i·7-s + 24.6i·8-s + 9·9-s − 36.1·10-s − 15.0i·11-s + 7.47·12-s − 23.7·14-s + 46.1i·15-s − 37.8·16-s − 90.8·17-s + 21.1i·18-s + ⋯
 L(s)  = 1 + 0.829i·2-s + 0.577·3-s + 0.311·4-s + 1.37i·5-s + 0.479i·6-s + 0.547i·7-s + 1.08i·8-s + 0.333·9-s − 1.14·10-s − 0.412i·11-s + 0.179·12-s − 0.453·14-s + 0.795i·15-s − 0.591·16-s − 1.29·17-s + 0.276i·18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $-0.999 + 0.0304i$ Analytic conductor: $$29.9139$$ Root analytic conductor: $$5.46936$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{507} (337, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 507,\ (\ :3/2),\ -0.999 + 0.0304i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.394448957$$ $$L(\frac12)$$ $$\approx$$ $$2.394448957$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 3T$$
13 $$1$$
good2 $$1 - 2.34iT - 8T^{2}$$
5 $$1 - 15.3iT - 125T^{2}$$
7 $$1 - 10.1iT - 343T^{2}$$
11 $$1 + 15.0iT - 1.33e3T^{2}$$
17 $$1 + 90.8T + 4.91e3T^{2}$$
19 $$1 - 114. iT - 6.85e3T^{2}$$
23 $$1 + 75.7T + 1.21e4T^{2}$$
29 $$1 - 214.T + 2.43e4T^{2}$$
31 $$1 + 284. iT - 2.97e4T^{2}$$
37 $$1 + 358. iT - 5.06e4T^{2}$$
41 $$1 - 313. iT - 6.89e4T^{2}$$
43 $$1 - 296.T + 7.95e4T^{2}$$
47 $$1 - 316. iT - 1.03e5T^{2}$$
53 $$1 - 163.T + 1.48e5T^{2}$$
59 $$1 + 254. iT - 2.05e5T^{2}$$
61 $$1 + 935.T + 2.26e5T^{2}$$
67 $$1 + 240. iT - 3.00e5T^{2}$$
71 $$1 - 947. iT - 3.57e5T^{2}$$
73 $$1 - 430. iT - 3.89e5T^{2}$$
79 $$1 + 496.T + 4.93e5T^{2}$$
83 $$1 - 392. iT - 5.71e5T^{2}$$
89 $$1 + 979. iT - 7.04e5T^{2}$$
97 $$1 + 553. iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$