Properties

 Label 2-12e3-8.3-c2-0-55 Degree $2$ Conductor $1728$ Sign $-0.258 + 0.965i$ Analytic cond. $47.0845$ Root an. cond. $6.86182$ Motivic weight $2$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 2.64i·5-s − 8.30i·7-s + 18.7·11-s − 22.3i·13-s + 23.4·17-s − 9.93·19-s − 32.3i·23-s + 18.0·25-s + 8.45i·29-s + 46.9i·31-s − 21.9·35-s + 28.3i·37-s − 77.7·41-s + 58.4·43-s − 54.2i·47-s + ⋯
 L(s)  = 1 − 0.528i·5-s − 1.18i·7-s + 1.70·11-s − 1.71i·13-s + 1.38·17-s − 0.522·19-s − 1.40i·23-s + 0.721·25-s + 0.291i·29-s + 1.51i·31-s − 0.626·35-s + 0.765i·37-s − 1.89·41-s + 1.35·43-s − 1.15i·47-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$1728$$    =    $$2^{6} \cdot 3^{3}$$ Sign: $-0.258 + 0.965i$ Analytic conductor: $$47.0845$$ Root analytic conductor: $$6.86182$$ Motivic weight: $$2$$ Rational: no Arithmetic: yes Character: $\chi_{1728} (1567, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 1728,\ (\ :1),\ -0.258 + 0.965i)$$

Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$2.306390650$$ $$L(\frac12)$$ $$\approx$$ $$2.306390650$$ $$L(2)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1$$
good5 $$1 + 2.64iT - 25T^{2}$$
7 $$1 + 8.30iT - 49T^{2}$$
11 $$1 - 18.7T + 121T^{2}$$
13 $$1 + 22.3iT - 169T^{2}$$
17 $$1 - 23.4T + 289T^{2}$$
19 $$1 + 9.93T + 361T^{2}$$
23 $$1 + 32.3iT - 529T^{2}$$
29 $$1 - 8.45iT - 841T^{2}$$
31 $$1 - 46.9iT - 961T^{2}$$
37 $$1 - 28.3iT - 1.36e3T^{2}$$
41 $$1 + 77.7T + 1.68e3T^{2}$$
43 $$1 - 58.4T + 1.84e3T^{2}$$
47 $$1 + 54.2iT - 2.20e3T^{2}$$
53 $$1 - 5.81iT - 2.80e3T^{2}$$
59 $$1 - 47.5T + 3.48e3T^{2}$$
61 $$1 - 27.7iT - 3.72e3T^{2}$$
67 $$1 - 50.6T + 4.48e3T^{2}$$
71 $$1 - 7.34iT - 5.04e3T^{2}$$
73 $$1 + 29.4T + 5.32e3T^{2}$$
79 $$1 - 43.2iT - 6.24e3T^{2}$$
83 $$1 - 10.7T + 6.88e3T^{2}$$
89 $$1 + 136.T + 7.92e3T^{2}$$
97 $$1 + 54.6T + 9.40e3T^{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

−8.630423435213986292617503739903, −8.364175094302590080326431582140, −7.18873983759873660986255388292, −6.67664248092793908317552662596, −5.59478175404000878877868242348, −4.73395943405462189137644330615, −3.82732921590061544465423974747, −3.08457062560276410929312712707, −1.30807121282824416229010146829, −0.70787331457370526183781694556, 1.39492309178523444560823938704, 2.30092928812064174152568026588, 3.51916317472474188957775383994, 4.23317220923141269883934193219, 5.46767001822174680504270273995, 6.22582967447228277211201362696, 6.84034092261129148878168231825, 7.73928203024959330650545072964, 8.845057345071624744833402509422, 9.288812182993928104118926901814