# Properties

 Label 2-3360-105.44-c0-0-0 Degree $2$ Conductor $3360$ Sign $-0.0633 - 0.997i$ Analytic cond. $1.67685$ Root an. cond. $1.29493$ Motivic weight $0$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Learn more

## Dirichlet series

 L(s)  = 1 + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)5-s + (−0.258 + 0.965i)7-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)15-s − i·21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (−0.707 + 0.707i)27-s − 1.73i·29-s + (0.707 − 0.707i)35-s + i·41-s + 1.93i·43-s − 45-s + (−0.707 + 1.22i)47-s + ⋯
 L(s)  = 1 + (−0.965 + 0.258i)3-s + (−0.866 − 0.5i)5-s + (−0.258 + 0.965i)7-s + (0.866 − 0.499i)9-s + (0.965 + 0.258i)15-s − i·21-s + (0.258 − 0.448i)23-s + (0.499 + 0.866i)25-s + (−0.707 + 0.707i)27-s − 1.73i·29-s + (0.707 − 0.707i)35-s + i·41-s + 1.93i·43-s − 45-s + (−0.707 + 1.22i)47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$3360$$    =    $$2^{5} \cdot 3 \cdot 5 \cdot 7$$ Sign: $-0.0633 - 0.997i$ Analytic conductor: $$1.67685$$ Root analytic conductor: $$1.29493$$ Motivic weight: $$0$$ Rational: no Arithmetic: yes Character: $\chi_{3360} (1409, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 3360,\ (\ :0),\ -0.0633 - 0.997i)$$

## Particular Values

 $$L(\frac{1}{2})$$ $$\approx$$ $$0.5282076591$$ $$L(\frac12)$$ $$\approx$$ $$0.5282076591$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (0.965 - 0.258i)T$$
5 $$1 + (0.866 + 0.5i)T$$
7 $$1 + (0.258 - 0.965i)T$$
good11 $$1 + (0.5 - 0.866i)T^{2}$$
13 $$1 - T^{2}$$
17 $$1 + (-0.5 + 0.866i)T^{2}$$
19 $$1 + (-0.5 - 0.866i)T^{2}$$
23 $$1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2}$$
29 $$1 + 1.73iT - T^{2}$$
31 $$1 + (-0.5 + 0.866i)T^{2}$$
37 $$1 + (0.5 + 0.866i)T^{2}$$
41 $$1 - iT - T^{2}$$
43 $$1 - 1.93iT - T^{2}$$
47 $$1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2}$$
53 $$1 + (-0.5 + 0.866i)T^{2}$$
59 $$1 + (0.5 - 0.866i)T^{2}$$
61 $$1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2}$$
67 $$1 + (-0.448 + 0.258i)T + (0.5 - 0.866i)T^{2}$$
71 $$1 - T^{2}$$
73 $$1 + (0.5 - 0.866i)T^{2}$$
79 $$1 + (-0.5 - 0.866i)T^{2}$$
83 $$1 - 1.93T + T^{2}$$
89 $$1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2}$$
97 $$1 - T^{2}$$
show more
show less
$$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

−9.172189863027620483295488421134, −8.128225169620690005434492580106, −7.64945071771355307949971915025, −6.44188088405126942790818291407, −6.10907006908632004938003840101, −5.05154207822042076312803723292, −4.58144835350433611549272172053, −3.67468599679942541534562726457, −2.60234222677032192678699537807, −1.11800789827827589191487110764, 0.42420924918273687271557559479, 1.76384034667518348539089468111, 3.28087587167552501671153106500, 3.90001133185999087651566928571, 4.81241458089678872255578060959, 5.54318744237412182128772267587, 6.69915725865853952294020853192, 6.96825607112059202061029299741, 7.60585667502200229751105431485, 8.427742701283115090504833289016