# Properties

 Label 2-2e5-32.29-c1-0-2 Degree $2$ Conductor $32$ Sign $0.555 + 0.831i$ Analytic cond. $0.255521$ Root an. cond. $0.505491$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.41i·2-s + (−0.707 − 0.292i)3-s − 2.00·4-s + (1.12 + 2.70i)5-s + (−0.414 + 1.00i)6-s + (1 − i)7-s + 2.82i·8-s + (−1.70 − 1.70i)9-s + (3.82 − 1.58i)10-s + (−4.12 + 1.70i)11-s + (1.41 + 0.585i)12-s + (0.292 − 0.707i)13-s + (−1.41 − 1.41i)14-s − 2.24i·15-s + 4.00·16-s − 2.82i·17-s + ⋯
 L(s)  = 1 − 0.999i·2-s + (−0.408 − 0.169i)3-s − 1.00·4-s + (0.501 + 1.21i)5-s + (−0.169 + 0.408i)6-s + (0.377 − 0.377i)7-s + 1.00i·8-s + (−0.569 − 0.569i)9-s + (1.21 − 0.501i)10-s + (−1.24 + 0.514i)11-s + (0.408 + 0.169i)12-s + (0.0812 − 0.196i)13-s + (−0.377 − 0.377i)14-s − 0.579i·15-s + 1.00·16-s − 0.685i·17-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$32$$    =    $$2^{5}$$ Sign: $0.555 + 0.831i$ Analytic conductor: $$0.255521$$ Root analytic conductor: $$0.505491$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{32} (29, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 32,\ (\ :1/2),\ 0.555 + 0.831i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.556099 - 0.297241i$$ $$L(\frac12)$$ $$\approx$$ $$0.556099 - 0.297241i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + 1.41iT$$
good3 $$1 + (0.707 + 0.292i)T + (2.12 + 2.12i)T^{2}$$
5 $$1 + (-1.12 - 2.70i)T + (-3.53 + 3.53i)T^{2}$$
7 $$1 + (-1 + i)T - 7iT^{2}$$
11 $$1 + (4.12 - 1.70i)T + (7.77 - 7.77i)T^{2}$$
13 $$1 + (-0.292 + 0.707i)T + (-9.19 - 9.19i)T^{2}$$
17 $$1 + 2.82iT - 17T^{2}$$
19 $$1 + (-1.53 + 3.70i)T + (-13.4 - 13.4i)T^{2}$$
23 $$1 + (-5.82 - 5.82i)T + 23iT^{2}$$
29 $$1 + (3.12 + 1.29i)T + (20.5 + 20.5i)T^{2}$$
31 $$1 + 4T + 31T^{2}$$
37 $$1 + (-0.292 - 0.707i)T + (-26.1 + 26.1i)T^{2}$$
41 $$1 + (0.171 + 0.171i)T + 41iT^{2}$$
43 $$1 + (-4.70 + 1.94i)T + (30.4 - 30.4i)T^{2}$$
47 $$1 + 0.343iT - 47T^{2}$$
53 $$1 + (1.12 - 0.464i)T + (37.4 - 37.4i)T^{2}$$
59 $$1 + (1.87 + 4.53i)T + (-41.7 + 41.7i)T^{2}$$
61 $$1 + (-1.70 - 0.707i)T + (43.1 + 43.1i)T^{2}$$
67 $$1 + (5.53 + 2.29i)T + (47.3 + 47.3i)T^{2}$$
71 $$1 + (5.82 - 5.82i)T - 71iT^{2}$$
73 $$1 + (-7 - 7i)T + 73iT^{2}$$
79 $$1 - 6iT - 79T^{2}$$
83 $$1 + (-1.87 + 4.53i)T + (-58.6 - 58.6i)T^{2}$$
89 $$1 + (-8.65 + 8.65i)T - 89iT^{2}$$
97 $$1 + 18.4T + 97T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$