# Properties

 Label 2-2888-152.43-c0-0-0 Degree $2$ Conductor $2888$ Sign $-0.600 - 0.799i$ Analytic cond. $1.44129$ Root an. cond. $1.20054$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.500 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.173 − 0.984i)16-s + (−1.87 + 0.684i)17-s + (−0.173 + 0.984i)22-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)32-s + (0.766 + 0.642i)33-s + (1.53 − 1.28i)34-s + ⋯
 L(s)  = 1 + (−0.939 + 0.342i)2-s + (−0.173 + 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.173 − 0.984i)6-s + (−0.500 + 0.866i)8-s + (0.5 − 0.866i)11-s + (0.5 + 0.866i)12-s + (0.173 − 0.984i)16-s + (−1.87 + 0.684i)17-s + (−0.173 + 0.984i)22-s + (−0.766 − 0.642i)24-s + (0.173 + 0.984i)25-s + (−0.5 + 0.866i)27-s + (0.173 + 0.984i)32-s + (0.766 + 0.642i)33-s + (1.53 − 1.28i)34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2888$$    =    $$2^{3} \cdot 19^{2}$$ Sign: $-0.600 - 0.799i$ Analytic conductor: $$1.44129$$ Root analytic conductor: $$1.20054$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2888} (1867, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2888,\ (\ :0),\ -0.600 - 0.799i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.6649874703$$ $$L(\frac12)$$ $$\approx$$ $$0.6649874703$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.939 - 0.342i)T$$
19 $$1$$
good3 $$1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}$$
5 $$1 + (-0.173 - 0.984i)T^{2}$$
7 $$1 + (0.5 - 0.866i)T^{2}$$
11 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
13 $$1 + (0.939 - 0.342i)T^{2}$$
17 $$1 + (1.87 - 0.684i)T + (0.766 - 0.642i)T^{2}$$
23 $$1 + (-0.173 + 0.984i)T^{2}$$
29 $$1 + (-0.766 - 0.642i)T^{2}$$
31 $$1 + (0.5 - 0.866i)T^{2}$$
37 $$1 - T^{2}$$
41 $$1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}$$
43 $$1 + (-1.53 - 1.28i)T + (0.173 + 0.984i)T^{2}$$
47 $$1 + (-0.766 - 0.642i)T^{2}$$
53 $$1 + (-0.173 + 0.984i)T^{2}$$
59 $$1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2}$$
61 $$1 + (-0.173 + 0.984i)T^{2}$$
67 $$1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2}$$
71 $$1 + (-0.173 - 0.984i)T^{2}$$
73 $$1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2}$$
79 $$1 + (0.939 + 0.342i)T^{2}$$
83 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
89 $$1 + (-0.347 - 1.96i)T + (-0.939 + 0.342i)T^{2}$$
97 $$1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)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

−9.282853918995797156868021930839, −8.625721937804754221439754983364, −7.912300848932111071654926253697, −6.90267644255066307415390916447, −6.30783012334885212419549940503, −5.48519093779802920444481530902, −4.58378914598783496287340410022, −3.75739115220094840020928697475, −2.58776253955292085717042848389, −1.34219073662482448052155765958, 0.60910059503852922791290117459, 1.95828242719723953100016845398, 2.36719589324225337726965757802, 3.82228519154998574299724434764, 4.65551833989971582584397563549, 6.04900231678502745651434416881, 6.85078991123578969011660908325, 7.06092673917340485063895094668, 7.891914451069816741927300565796, 8.804620283198922509362667604369