# Properties

 Label 2-880-11.3-c1-0-8 Degree $2$ Conductor $880$ Sign $0.999 + 0.0206i$ Analytic cond. $7.02683$ Root an. cond. $2.65081$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.529 − 1.62i)3-s + (0.809 + 0.587i)5-s + (−1.14 + 3.52i)7-s + (0.0536 − 0.0389i)9-s + (−2.68 − 1.94i)11-s + (0.952 − 0.692i)13-s + (0.529 − 1.62i)15-s + (4.36 + 3.17i)17-s + (1.18 + 3.65i)19-s + 6.34·21-s + 8.68·23-s + (0.309 + 0.951i)25-s + (−4.24 − 3.08i)27-s + (−2.12 + 6.53i)29-s + (7.08 − 5.14i)31-s + ⋯
 L(s)  = 1 + (−0.305 − 0.940i)3-s + (0.361 + 0.262i)5-s + (−0.432 + 1.33i)7-s + (0.0178 − 0.0129i)9-s + (−0.810 − 0.585i)11-s + (0.264 − 0.191i)13-s + (0.136 − 0.420i)15-s + (1.05 + 0.769i)17-s + (0.272 + 0.839i)19-s + 1.38·21-s + 1.81·23-s + (0.0618 + 0.190i)25-s + (−0.817 − 0.594i)27-s + (−0.394 + 1.21i)29-s + (1.27 − 0.924i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0206i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$880$$    =    $$2^{4} \cdot 5 \cdot 11$$ Sign: $0.999 + 0.0206i$ Analytic conductor: $$7.02683$$ Root analytic conductor: $$2.65081$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{880} (641, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 880,\ (\ :1/2),\ 0.999 + 0.0206i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.46702 - 0.0151531i$$ $$L(\frac12)$$ $$\approx$$ $$1.46702 - 0.0151531i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
5 $$1 + (-0.809 - 0.587i)T$$
11 $$1 + (2.68 + 1.94i)T$$
good3 $$1 + (0.529 + 1.62i)T + (-2.42 + 1.76i)T^{2}$$
7 $$1 + (1.14 - 3.52i)T + (-5.66 - 4.11i)T^{2}$$
13 $$1 + (-0.952 + 0.692i)T + (4.01 - 12.3i)T^{2}$$
17 $$1 + (-4.36 - 3.17i)T + (5.25 + 16.1i)T^{2}$$
19 $$1 + (-1.18 - 3.65i)T + (-15.3 + 11.1i)T^{2}$$
23 $$1 - 8.68T + 23T^{2}$$
29 $$1 + (2.12 - 6.53i)T + (-23.4 - 17.0i)T^{2}$$
31 $$1 + (-7.08 + 5.14i)T + (9.57 - 29.4i)T^{2}$$
37 $$1 + (-0.696 + 2.14i)T + (-29.9 - 21.7i)T^{2}$$
41 $$1 + (-0.493 - 1.51i)T + (-33.1 + 24.0i)T^{2}$$
43 $$1 + 4.11T + 43T^{2}$$
47 $$1 + (-3.91 - 12.0i)T + (-38.0 + 27.6i)T^{2}$$
53 $$1 + (-10.0 + 7.27i)T + (16.3 - 50.4i)T^{2}$$
59 $$1 + (-0.121 + 0.374i)T + (-47.7 - 34.6i)T^{2}$$
61 $$1 + (1.45 + 1.05i)T + (18.8 + 58.0i)T^{2}$$
67 $$1 - 14.3T + 67T^{2}$$
71 $$1 + (5.54 + 4.02i)T + (21.9 + 67.5i)T^{2}$$
73 $$1 + (-3.10 + 9.56i)T + (-59.0 - 42.9i)T^{2}$$
79 $$1 + (0.901 - 0.654i)T + (24.4 - 75.1i)T^{2}$$
83 $$1 + (-0.0140 - 0.0101i)T + (25.6 + 78.9i)T^{2}$$
89 $$1 + 8.49T + 89T^{2}$$
97 $$1 + (8.50 - 6.18i)T + (29.9 - 92.2i)T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−10.12928589997160058287104293080, −9.279739187375656034973836881718, −8.347298224645001258267515195103, −7.59424945243083614328021437332, −6.53321568154535794901707201465, −5.86282266250004501275117678070, −5.27873287334432058356071670836, −3.43291701645815627572933881042, −2.52854666005754416282488818944, −1.19565574960920464737606746182, 0.911416885261561487896070151605, 2.81207590913558847250912048115, 3.96195757863562109447506473000, 4.83868485731211605093477501688, 5.42851477283673468607801903771, 6.92560663873292625204511944421, 7.35753528003260948837647022914, 8.613663049774648429577697359287, 9.771342812357913233112602303310, 9.977594936828869547794769026406