# Properties

 Label 2-507-13.12-c3-0-56 Degree $2$ Conductor $507$ Sign $-0.277 + 0.960i$ 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 − 3·3-s + 8·4-s − 5.19i·5-s + 10.3i·7-s + 9·9-s − 51.9i·11-s − 24·12-s + 15.5i·15-s + 64·16-s − 117·17-s − 24.2i·19-s − 41.5i·20-s − 31.1i·21-s + 18·23-s + 98·25-s + ⋯
 L(s)  = 1 − 0.577·3-s + 4-s − 0.464i·5-s + 0.561i·7-s + 0.333·9-s − 1.42i·11-s − 0.577·12-s + 0.268i·15-s + 16-s − 1.66·17-s − 0.292i·19-s − 0.464i·20-s − 0.323i·21-s + 0.163·23-s + 0.784·25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $-0.277 + 0.960i$ 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: $$(2,\ 507,\ (\ :3/2),\ -0.277 + 0.960i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.450193101$$ $$L(\frac12)$$ $$\approx$$ $$1.450193101$$ $$L(\frac{5}{2})$$ 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 - 8T^{2}$$
5 $$1 + 5.19iT - 125T^{2}$$
7 $$1 - 10.3iT - 343T^{2}$$
11 $$1 + 51.9iT - 1.33e3T^{2}$$
17 $$1 + 117T + 4.91e3T^{2}$$
19 $$1 + 24.2iT - 6.85e3T^{2}$$
23 $$1 - 18T + 1.21e4T^{2}$$
29 $$1 + 99T + 2.43e4T^{2}$$
31 $$1 + 193. iT - 2.97e4T^{2}$$
37 $$1 + 112. iT - 5.06e4T^{2}$$
41 $$1 - 36.3iT - 6.89e4T^{2}$$
43 $$1 + 82T + 7.95e4T^{2}$$
47 $$1 - 72.7iT - 1.03e5T^{2}$$
53 $$1 + 261T + 1.48e5T^{2}$$
59 $$1 + 789. iT - 2.05e5T^{2}$$
61 $$1 + 719T + 2.26e5T^{2}$$
67 $$1 - 703. iT - 3.00e5T^{2}$$
71 $$1 + 467. iT - 3.57e5T^{2}$$
73 $$1 + 684. iT - 3.89e5T^{2}$$
79 $$1 + 440T + 4.93e5T^{2}$$
83 $$1 + 1.19e3iT - 5.71e5T^{2}$$
89 $$1 + 1.51e3iT - 7.04e5T^{2}$$
97 $$1 + 1.15e3iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$