# Properties

 Label 2-3920-20.19-c0-0-6 Degree $2$ Conductor $3920$ Sign $0.866 - 0.5i$ Analytic cond. $1.95633$ Root an. cond. $1.39869$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.73·3-s + i·5-s + 1.99·9-s − 1.73i·11-s + i·13-s + 1.73i·15-s + i·17-s − 25-s + 1.73·27-s + 29-s − 2.99i·33-s + 1.73i·39-s + 1.99i·45-s − 1.73·47-s + 1.73i·51-s + ⋯
 L(s)  = 1 + 1.73·3-s + i·5-s + 1.99·9-s − 1.73i·11-s + i·13-s + 1.73i·15-s + i·17-s − 25-s + 1.73·27-s + 29-s − 2.99i·33-s + 1.73i·39-s + 1.99i·45-s − 1.73·47-s + 1.73i·51-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.5i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3920$$    =    $$2^{4} \cdot 5 \cdot 7^{2}$$ Sign: $0.866 - 0.5i$ Analytic conductor: $$1.95633$$ Root analytic conductor: $$1.39869$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3920} (3039, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3920,\ (\ :0),\ 0.866 - 0.5i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$2.425203920$$ $$L(\frac12)$$ $$\approx$$ $$2.425203920$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

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

−8.620934144896486810533805281542, −8.116309383987843399647398539288, −7.45107028408182476548797906625, −6.51703713663024606094925999794, −6.09975052674398827965950187913, −4.66629496976202453060037807522, −3.64394913070110952399276718580, −3.33550101485453386116903008289, −2.48917894556172260893005336881, −1.59646771079412109620937701749, 1.29618632104580819877361630414, 2.24979900847689935477682389354, 2.97064759651776962625400815123, 3.97851471962543880594130770281, 4.68206754014833178583060037693, 5.28886286155025898388959890409, 6.68902735479707347572189483672, 7.41544422845730055219745958040, 7.978018907108165640814119507960, 8.537218119203944927726220286376