# Properties

 Label 2-80-80.67-c1-0-9 Degree $2$ Conductor $80$ Sign $-0.581 + 0.813i$ Analytic cond. $0.638803$ Root an. cond. $0.799251$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.307 − 1.38i)2-s − 2.85i·3-s + (−1.81 + 0.849i)4-s + (1.43 + 1.71i)5-s + (−3.94 + 0.879i)6-s + (−0.458 − 0.458i)7-s + (1.73 + 2.23i)8-s − 5.15·9-s + (1.92 − 2.50i)10-s + (−0.492 − 0.492i)11-s + (2.42 + 5.17i)12-s + 4.52·13-s + (−0.492 + 0.774i)14-s + (4.89 − 4.09i)15-s + (2.55 − 3.07i)16-s + (−3.12 − 3.12i)17-s + ⋯
 L(s)  = 1 + (−0.217 − 0.976i)2-s − 1.64i·3-s + (−0.905 + 0.424i)4-s + (0.641 + 0.766i)5-s + (−1.60 + 0.358i)6-s + (−0.173 − 0.173i)7-s + (0.611 + 0.791i)8-s − 1.71·9-s + (0.608 − 0.793i)10-s + (−0.148 − 0.148i)11-s + (0.700 + 1.49i)12-s + 1.25·13-s + (−0.131 + 0.207i)14-s + (1.26 − 1.05i)15-s + (0.638 − 0.769i)16-s + (−0.758 − 0.758i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$80$$    =    $$2^{4} \cdot 5$$ Sign: $-0.581 + 0.813i$ Analytic conductor: $$0.638803$$ Root analytic conductor: $$0.799251$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{80} (67, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 80,\ (\ :1/2),\ -0.581 + 0.813i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.390367 - 0.759155i$$ $$L(\frac12)$$ $$\approx$$ $$0.390367 - 0.759155i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.307 + 1.38i)T$$
5 $$1 + (-1.43 - 1.71i)T$$
good3 $$1 + 2.85iT - 3T^{2}$$
7 $$1 + (0.458 + 0.458i)T + 7iT^{2}$$
11 $$1 + (0.492 + 0.492i)T + 11iT^{2}$$
13 $$1 - 4.52T + 13T^{2}$$
17 $$1 + (3.12 + 3.12i)T + 17iT^{2}$$
19 $$1 + (-4.04 - 4.04i)T + 19iT^{2}$$
23 $$1 + (1.80 - 1.80i)T - 23iT^{2}$$
29 $$1 + (3.83 - 3.83i)T - 29iT^{2}$$
31 $$1 - 0.139iT - 31T^{2}$$
37 $$1 - 5.84T + 37T^{2}$$
41 $$1 - 4.55iT - 41T^{2}$$
43 $$1 + 7.49T + 43T^{2}$$
47 $$1 + (4.14 - 4.14i)T - 47iT^{2}$$
53 $$1 + 2.75iT - 53T^{2}$$
59 $$1 + (3.62 - 3.62i)T - 59iT^{2}$$
61 $$1 + (-3.72 - 3.72i)T + 61iT^{2}$$
67 $$1 - 3.32T + 67T^{2}$$
71 $$1 - 1.37T + 71T^{2}$$
73 $$1 + (2.55 + 2.55i)T + 73iT^{2}$$
79 $$1 + 3.86T + 79T^{2}$$
83 $$1 + 14.4iT - 83T^{2}$$
89 $$1 + 3.35T + 89T^{2}$$
97 $$1 + (4.95 + 4.95i)T + 97iT^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$