# Properties

 Label 2-507-39.20-c1-0-16 Degree $2$ Conductor $507$ Sign $-0.430 - 0.902i$ 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 + (0.389 + 1.45i)2-s + (0.239 + 1.71i)3-s + (−0.232 + 0.133i)4-s + (1.06 − 1.06i)5-s + (−2.40 + 1.01i)6-s + (1.36 + 0.366i)7-s + (1.84 + 1.84i)8-s + (−2.88 + 0.820i)9-s + (1.96 + 1.13i)10-s + (3.97 − 1.06i)11-s + (−0.285 − 0.366i)12-s + 2.12i·14-s + (2.08 + 1.57i)15-s + (−2.23 + 3.86i)16-s + (−2.51 − 4.36i)17-s + (−2.31 − 3.87i)18-s + ⋯
 L(s)  = 1 + (0.275 + 1.02i)2-s + (0.138 + 0.990i)3-s + (−0.116 + 0.0669i)4-s + (0.476 − 0.476i)5-s + (−0.980 + 0.415i)6-s + (0.516 + 0.138i)7-s + (0.652 + 0.652i)8-s + (−0.961 + 0.273i)9-s + (0.621 + 0.358i)10-s + (1.19 − 0.321i)11-s + (−0.0823 − 0.105i)12-s + 0.569i·14-s + (0.537 + 0.405i)15-s + (−0.558 + 0.966i)16-s + (−0.611 − 1.05i)17-s + (−0.546 − 0.913i)18-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.430 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$507$$    =    $$3 \cdot 13^{2}$$ Sign: $-0.430 - 0.902i$ Analytic conductor: $$4.04841$$ Root analytic conductor: $$2.01206$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{507} (488, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 507,\ (\ :1/2),\ -0.430 - 0.902i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.15239 + 1.82706i$$ $$L(\frac12)$$ $$\approx$$ $$1.15239 + 1.82706i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1 + (-0.239 - 1.71i)T$$
13 $$1$$
good2 $$1 + (-0.389 - 1.45i)T + (-1.73 + i)T^{2}$$
5 $$1 + (-1.06 + 1.06i)T - 5iT^{2}$$
7 $$1 + (-1.36 - 0.366i)T + (6.06 + 3.5i)T^{2}$$
11 $$1 + (-3.97 + 1.06i)T + (9.52 - 5.5i)T^{2}$$
17 $$1 + (2.51 + 4.36i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (1 - 3.73i)T + (-16.4 - 9.5i)T^{2}$$
23 $$1 + (-11.5 - 19.9i)T^{2}$$
29 $$1 + (6.20 + 3.58i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (-2.46 - 2.46i)T + 31iT^{2}$$
37 $$1 + (1.40 + 5.23i)T + (-32.0 + 18.5i)T^{2}$$
41 $$1 + (1.45 + 5.42i)T + (-35.5 + 20.5i)T^{2}$$
43 $$1 + (1.90 - 1.09i)T + (21.5 - 37.2i)T^{2}$$
47 $$1 + (4.25 + 4.25i)T + 47iT^{2}$$
53 $$1 - 0.779iT - 53T^{2}$$
59 $$1 + (-0.779 + 2.90i)T + (-51.0 - 29.5i)T^{2}$$
61 $$1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2}$$
67 $$1 + (5.73 - 1.53i)T + (58.0 - 33.5i)T^{2}$$
71 $$1 + (-2.90 - 0.779i)T + (61.4 + 35.5i)T^{2}$$
73 $$1 + (-0.901 + 0.901i)T - 73iT^{2}$$
79 $$1 - 2T + 79T^{2}$$
83 $$1 + (-2.90 + 2.90i)T - 83iT^{2}$$
89 $$1 + (-9.01 + 2.41i)T + (77.0 - 44.5i)T^{2}$$
97 $$1 + (-0.437 + 1.63i)T + (-84.0 - 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$