# Properties

 Label 2-507-39.5-c1-0-25 Degree $2$ Conductor $507$ Sign $0.957 + 0.289i$ 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 + 1.73·3-s − 2i·4-s + (3.09 + 3.09i)7-s + 2.99·9-s − 3.46i·12-s − 4·16-s + (−2.26 + 2.26i)19-s + (5.36 + 5.36i)21-s − 5i·25-s + 5.19·27-s + (6.19 − 6.19i)28-s + (0.830 − 0.830i)31-s − 5.99i·36-s + (−8.46 − 8.46i)37-s + 1.73i·43-s + ⋯
 L(s)  = 1 + 1.00·3-s − i·4-s + (1.17 + 1.17i)7-s + 0.999·9-s − 0.999i·12-s − 16-s + (−0.520 + 0.520i)19-s + (1.17 + 1.17i)21-s − i·25-s + 1.00·27-s + (1.17 − 1.17i)28-s + (0.149 − 0.149i)31-s − 0.999i·36-s + (−1.39 − 1.39i)37-s + 0.264i·43-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $0.957 + 0.289i$ Analytic conductor: $$4.04841$$ Root analytic conductor: $$2.01206$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{507} (239, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 507,\ (\ :1/2),\ 0.957 + 0.289i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.17242 - 0.321668i$$ $$L(\frac12)$$ $$\approx$$ $$2.17242 - 0.321668i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 - 1.73T$$
13 $$1$$
good2 $$1 + 2iT^{2}$$
5 $$1 + 5iT^{2}$$
7 $$1 + (-3.09 - 3.09i)T + 7iT^{2}$$
11 $$1 - 11iT^{2}$$
17 $$1 + 17T^{2}$$
19 $$1 + (2.26 - 2.26i)T - 19iT^{2}$$
23 $$1 + 23T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + (-0.830 + 0.830i)T - 31iT^{2}$$
37 $$1 + (8.46 + 8.46i)T + 37iT^{2}$$
41 $$1 + 41iT^{2}$$
43 $$1 - 1.73iT - 43T^{2}$$
47 $$1 - 47iT^{2}$$
53 $$1 - 53T^{2}$$
59 $$1 - 59iT^{2}$$
61 $$1 - 8.66T + 61T^{2}$$
67 $$1 + (11.5 - 11.5i)T - 67iT^{2}$$
71 $$1 + 71iT^{2}$$
73 $$1 + (7.63 + 7.63i)T + 73iT^{2}$$
79 $$1 + 12.1T + 79T^{2}$$
83 $$1 + 83iT^{2}$$
89 $$1 - 89iT^{2}$$
97 $$1 + (7.02 - 7.02i)T - 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$