# Properties

 Label 2-2548-364.263-c0-0-0 Degree $2$ Conductor $2548$ Sign $0.128 - 0.991i$ Analytic cond. $1.27161$ Root an. cond. $1.12766$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s + 9-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.499 − 0.866i)20-s + 0.999·26-s + (0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s − 0.999·34-s + ⋯
 L(s)  = 1 + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + 0.999·8-s + 9-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 + 0.866i)16-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)18-s + (0.499 − 0.866i)20-s + 0.999·26-s + (0.5 + 0.866i)29-s + (−0.499 − 0.866i)32-s − 0.999·34-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 2548 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$2548$$    =    $$2^{2} \cdot 7^{2} \cdot 13$$ Sign: $0.128 - 0.991i$ Analytic conductor: $$1.27161$$ Root analytic conductor: $$1.12766$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{2548} (263, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 2548,\ (\ :0),\ 0.128 - 0.991i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.069973285$$ $$L(\frac12)$$ $$\approx$$ $$1.069973285$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + (0.5 - 0.866i)T$$
7 $$1$$
13 $$1 + (0.5 + 0.866i)T$$
good3 $$1 - T^{2}$$
5 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 - T^{2}$$
17 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
19 $$1 - T^{2}$$
23 $$1 + (0.5 + 0.866i)T^{2}$$
29 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
31 $$1 + (0.5 + 0.866i)T^{2}$$
37 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
41 $$1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 + 0.866i)T^{2}$$
47 $$1 + (0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + T + T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + (0.5 + 0.866i)T^{2}$$
73 $$1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2}$$
79 $$1 + (0.5 - 0.866i)T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2}$$
97 $$1 + (1 - 1.73i)T + (-0.5 - 0.866i)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

−9.372759250997129757723669293646, −8.280870944479071571542216532270, −7.69689779615188450859645286479, −6.95612315131344762548278446431, −6.36118402844847364745752330287, −5.58449089320518232905719832117, −4.74387043071975521138624765837, −3.71295729323798562955640664206, −2.47759962026846324014870392190, −1.27285184520674591537394984693, 1.03728765299495148164202882245, 1.92715743686348076457171395992, 2.94770513097684583366159477809, 4.24415461387670700211252450772, 4.63577592817786342317343909150, 5.61624032167894390090320536363, 6.88521173603733345270066226813, 7.48895299926007825561392559393, 8.370394961894453363948651735784, 9.169994380759041792596594353303