# Properties

 Degree $2$ Conductor $3528$ Sign $0.458 - 0.888i$ Motivic weight $0$ Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − i·2-s + (−0.258 + 0.965i)3-s − 4-s + (0.965 + 1.67i)5-s + (0.965 + 0.258i)6-s + i·8-s + (−0.866 − 0.499i)9-s + (1.67 − 0.965i)10-s + (0.258 − 0.965i)12-s + (1.22 + 0.707i)13-s + (−1.86 + 0.500i)15-s + 16-s + (−0.499 + 0.866i)18-s + (0.448 + 0.258i)19-s + (−0.965 − 1.67i)20-s + ⋯
 L(s)  = 1 − i·2-s + (−0.258 + 0.965i)3-s − 4-s + (0.965 + 1.67i)5-s + (0.965 + 0.258i)6-s + i·8-s + (−0.866 − 0.499i)9-s + (1.67 − 0.965i)10-s + (0.258 − 0.965i)12-s + (1.22 + 0.707i)13-s + (−1.86 + 0.500i)15-s + 16-s + (−0.499 + 0.866i)18-s + (0.448 + 0.258i)19-s + (−0.965 − 1.67i)20-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.458 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3528$$    =    $$2^{3} \cdot 3^{2} \cdot 7^{2}$$ Sign: $0.458 - 0.888i$ Motivic weight: $$0$$ Character: $\chi_{3528} (1157, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3528,\ (\ :0),\ 0.458 - 0.888i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$1.286612244$$ $$L(\frac12)$$ $$\approx$$ $$1.286612244$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1 + iT$$
3 $$1 + (0.258 - 0.965i)T$$
7 $$1$$
good5 $$1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2}$$
11 $$1 + (-0.5 - 0.866i)T^{2}$$
13 $$1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2}$$
17 $$1 + (0.5 - 0.866i)T^{2}$$
19 $$1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2}$$
23 $$1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2}$$
29 $$1 + (-0.5 + 0.866i)T^{2}$$
31 $$1 + T^{2}$$
37 $$1 + (0.5 + 0.866i)T^{2}$$
41 $$1 + (0.5 + 0.866i)T^{2}$$
43 $$1 + (0.5 - 0.866i)T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + (-0.5 + 0.866i)T^{2}$$
59 $$1 + 1.41T + T^{2}$$
61 $$1 - 0.517iT - T^{2}$$
67 $$1 - T^{2}$$
71 $$1 + iT - T^{2}$$
73 $$1 + (-0.5 + 0.866i)T^{2}$$
79 $$1 + 1.73T + T^{2}$$
83 $$1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2}$$
89 $$1 + (0.5 + 0.866i)T^{2}$$
97 $$1 + (-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

−9.165577967771647470969433990551, −8.581818557452514645912630253593, −7.32031426074496253247137809393, −6.36871653621315665942414871855, −5.88010399070666740681404549541, −4.96846847574743718275150662681, −4.01484396126455789669258825486, −3.21449256421386131795569253312, −2.72664569764473842005391873648, −1.54520793365046715443503950040, 0.907058511574132158144474429475, 1.50218452842280610816521720598, 3.08521346697873599964414926152, 4.38455837995271110714839453538, 5.32753022457308140538263666776, 5.54373084809328130077444053959, 6.27848195622828691187989616760, 7.09288210279048995425797996933, 7.978134683860525229908423003857, 8.492859760754661392526445596426