# Properties

 Label 2-507-13.12-c3-0-37 Degree $2$ Conductor $507$ Sign $0.832 - 0.554i$ 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 + 12i·5-s + 2i·7-s + 9·9-s − 36i·11-s − 24·12-s − 36i·15-s + 64·16-s + 78·17-s − 74i·19-s + 96i·20-s − 6i·21-s + 96·23-s − 19·25-s + ⋯
 L(s)  = 1 − 0.577·3-s + 4-s + 1.07i·5-s + 0.107i·7-s + 0.333·9-s − 0.986i·11-s − 0.577·12-s − 0.619i·15-s + 16-s + 1.11·17-s − 0.893i·19-s + 1.07i·20-s − 0.0623i·21-s + 0.870·23-s − 0.151·25-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $0.832 - 0.554i$ 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: $$(2,\ 507,\ (\ :3/2),\ 0.832 - 0.554i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$2.214337380$$ $$L(\frac12)$$ $$\approx$$ $$2.214337380$$ $$L(\frac{5}{2})$$ 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 - 12iT - 125T^{2}$$
7 $$1 - 2iT - 343T^{2}$$
11 $$1 + 36iT - 1.33e3T^{2}$$
17 $$1 - 78T + 4.91e3T^{2}$$
19 $$1 + 74iT - 6.85e3T^{2}$$
23 $$1 - 96T + 1.21e4T^{2}$$
29 $$1 - 18T + 2.43e4T^{2}$$
31 $$1 - 214iT - 2.97e4T^{2}$$
37 $$1 + 286iT - 5.06e4T^{2}$$
41 $$1 - 384iT - 6.89e4T^{2}$$
43 $$1 + 524T + 7.95e4T^{2}$$
47 $$1 - 300iT - 1.03e5T^{2}$$
53 $$1 - 558T + 1.48e5T^{2}$$
59 $$1 - 576iT - 2.05e5T^{2}$$
61 $$1 - 74T + 2.26e5T^{2}$$
67 $$1 + 38iT - 3.00e5T^{2}$$
71 $$1 - 456iT - 3.57e5T^{2}$$
73 $$1 + 682iT - 3.89e5T^{2}$$
79 $$1 - 704T + 4.93e5T^{2}$$
83 $$1 - 888iT - 5.71e5T^{2}$$
89 $$1 + 1.02e3iT - 7.04e5T^{2}$$
97 $$1 + 110iT - 9.12e5T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$