# Properties

 Label 2-55-55.32-c1-0-0 Degree $2$ Conductor $55$ Sign $-0.969 - 0.244i$ Analytic cond. $0.439177$ Root an. cond. $0.662704$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.58 + 1.58i)2-s + (−1 + i)3-s − 3.00i·4-s + (−2 + i)5-s − 3.16i·6-s + (1.58 + 1.58i)8-s + i·9-s + (1.58 − 4.74i)10-s + (1 + 3.16i)11-s + (3.00 + 3.00i)12-s + (3.16 + 3.16i)13-s + (1 − 3i)15-s + 0.999·16-s + (3.16 − 3.16i)17-s + (−1.58 − 1.58i)18-s − 6.32·19-s + ⋯
 L(s)  = 1 + (−1.11 + 1.11i)2-s + (−0.577 + 0.577i)3-s − 1.50i·4-s + (−0.894 + 0.447i)5-s − 1.29i·6-s + (0.559 + 0.559i)8-s + 0.333i·9-s + (0.500 − 1.50i)10-s + (0.301 + 0.953i)11-s + (0.866 + 0.866i)12-s + (0.877 + 0.877i)13-s + (0.258 − 0.774i)15-s + 0.249·16-s + (0.766 − 0.766i)17-s + (−0.372 − 0.372i)18-s − 1.45·19-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.244i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$55$$    =    $$5 \cdot 11$$ Sign: $-0.969 - 0.244i$ Analytic conductor: $$0.439177$$ Root analytic conductor: $$0.662704$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{55} (32, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 55,\ (\ :1/2),\ -0.969 - 0.244i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.0433953 + 0.349167i$$ $$L(\frac12)$$ $$\approx$$ $$0.0433953 + 0.349167i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad5 $$1 + (2 - i)T$$
11 $$1 + (-1 - 3.16i)T$$
good2 $$1 + (1.58 - 1.58i)T - 2iT^{2}$$
3 $$1 + (1 - i)T - 3iT^{2}$$
7 $$1 - 7iT^{2}$$
13 $$1 + (-3.16 - 3.16i)T + 13iT^{2}$$
17 $$1 + (-3.16 + 3.16i)T - 17iT^{2}$$
19 $$1 + 6.32T + 19T^{2}$$
23 $$1 + (1 - i)T - 23iT^{2}$$
29 $$1 - 6.32T + 29T^{2}$$
31 $$1 - 2T + 31T^{2}$$
37 $$1 + (-3 - 3i)T + 37iT^{2}$$
41 $$1 + 6.32iT - 41T^{2}$$
43 $$1 + 43iT^{2}$$
47 $$1 + (-3 - 3i)T + 47iT^{2}$$
53 $$1 + (1 - i)T - 53iT^{2}$$
59 $$1 + 6iT - 59T^{2}$$
61 $$1 - 6.32iT - 61T^{2}$$
67 $$1 + (-3 - 3i)T + 67iT^{2}$$
71 $$1 + 8T + 71T^{2}$$
73 $$1 + (3.16 + 3.16i)T + 73iT^{2}$$
79 $$1 - 6.32T + 79T^{2}$$
83 $$1 + (-6.32 - 6.32i)T + 83iT^{2}$$
89 $$1 + 6iT - 89T^{2}$$
97 $$1 + (7 + 7i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$