# Properties

 Label 1-2e5-32.21-r0-0-0 Degree $1$ Conductor $32$ Sign $0.831 - 0.555i$ Analytic cond. $0.148607$ Root an. cond. $0.148607$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + i·7-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − 15-s − 17-s + (−0.707 + 0.707i)19-s + (0.707 + 0.707i)21-s − i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + 31-s + ⋯
 L(s)  = 1 + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)5-s + i·7-s − i·9-s + (0.707 + 0.707i)11-s + (−0.707 + 0.707i)13-s − 15-s − 17-s + (−0.707 + 0.707i)19-s + (0.707 + 0.707i)21-s − i·23-s + i·25-s + (−0.707 − 0.707i)27-s + (0.707 − 0.707i)29-s + 31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.831 - 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$1$$ Conductor: $$32$$    =    $$2^{5}$$ Sign: $0.831 - 0.555i$ Analytic conductor: $$0.148607$$ Root analytic conductor: $$0.148607$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{32} (21, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(1,\ 32,\ (0:\ ),\ 0.831 - 0.555i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.7969684494 - 0.2417577360i$$ $$L(\frac12)$$ $$\approx$$ $$0.7969684494 - 0.2417577360i$$ $$L(1)$$ $$\approx$$ $$1.012975691 - 0.2138733330i$$ $$L(1)$$ $$\approx$$ $$1.012975691 - 0.2138733330i$$

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + (0.707 - 0.707i)T$$
5 $$1 + (-0.707 - 0.707i)T$$
7 $$1 + iT$$
11 $$1 + (0.707 + 0.707i)T$$
13 $$1 + (-0.707 + 0.707i)T$$
17 $$1 - T$$
19 $$1 + (-0.707 + 0.707i)T$$
23 $$1 - iT$$
29 $$1 + (0.707 - 0.707i)T$$
31 $$1 + T$$
37 $$1 + (-0.707 - 0.707i)T$$
41 $$1 - iT$$
43 $$1 + (0.707 + 0.707i)T$$
47 $$1 - T$$
53 $$1 + (0.707 + 0.707i)T$$
59 $$1 + (-0.707 - 0.707i)T$$
61 $$1 + (0.707 - 0.707i)T$$
67 $$1 + (0.707 - 0.707i)T$$
71 $$1 + iT$$
73 $$1 - iT$$
79 $$1 - T$$
83 $$1 + (-0.707 + 0.707i)T$$
89 $$1 + iT$$
97 $$1 + T$$
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$