# Properties

 Label 2-462-33.32-c1-0-0 Degree $2$ Conductor $462$ Sign $-0.933 - 0.359i$ Analytic cond. $3.68908$ Root an. cond. $1.92070$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 2-s + (−0.742 + 1.56i)3-s + 4-s − 3.23i·5-s + (0.742 − 1.56i)6-s + i·7-s − 8-s + (−1.89 − 2.32i)9-s + 3.23i·10-s + (−3.30 + 0.248i)11-s + (−0.742 + 1.56i)12-s + 5.12i·13-s − i·14-s + (5.05 + 2.39i)15-s + 16-s − 2.90·17-s + ⋯
 L(s)  = 1 − 0.707·2-s + (−0.428 + 0.903i)3-s + 0.5·4-s − 1.44i·5-s + (0.303 − 0.638i)6-s + 0.377i·7-s − 0.353·8-s + (−0.632 − 0.774i)9-s + 1.02i·10-s + (−0.997 + 0.0749i)11-s + (−0.214 + 0.451i)12-s + 1.42i·13-s − 0.267i·14-s + (1.30 + 0.619i)15-s + 0.250·16-s − 0.703·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$462$$    =    $$2 \cdot 3 \cdot 7 \cdot 11$$ Sign: $-0.933 - 0.359i$ Analytic conductor: $$3.68908$$ Root analytic conductor: $$1.92070$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{462} (197, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 462,\ (\ :1/2),\ -0.933 - 0.359i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0480894 + 0.258397i$$ $$L(\frac12)$$ $$\approx$$ $$0.0480894 + 0.258397i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + T$$
3 $$1 + (0.742 - 1.56i)T$$
7 $$1 - iT$$
11 $$1 + (3.30 - 0.248i)T$$
good5 $$1 + 3.23iT - 5T^{2}$$
13 $$1 - 5.12iT - 13T^{2}$$
17 $$1 + 2.90T + 17T^{2}$$
19 $$1 - 0.590iT - 19T^{2}$$
23 $$1 - 9.26iT - 23T^{2}$$
29 $$1 + 0.996T + 29T^{2}$$
31 $$1 + 4.39T + 31T^{2}$$
37 $$1 + 8.10T + 37T^{2}$$
41 $$1 + 9.80T + 41T^{2}$$
43 $$1 - 1.94iT - 43T^{2}$$
47 $$1 + 7.19iT - 47T^{2}$$
53 $$1 - 0.204iT - 53T^{2}$$
59 $$1 - 6.09iT - 59T^{2}$$
61 $$1 + 5.31iT - 61T^{2}$$
67 $$1 - 8.79T + 67T^{2}$$
71 $$1 - 6.62iT - 71T^{2}$$
73 $$1 + 7.07iT - 73T^{2}$$
79 $$1 - 5.49iT - 79T^{2}$$
83 $$1 - 11.0T + 83T^{2}$$
89 $$1 + 1.28iT - 89T^{2}$$
97 $$1 + 11.3T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$