# Properties

 Label 2-432-108.11-c1-0-7 Degree $2$ Conductor $432$ Sign $0.459 - 0.887i$ Analytic cond. $3.44953$ Root an. cond. $1.85729$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 + (1.15 + 1.28i)3-s + (−0.152 + 0.417i)5-s + (2.02 + 0.357i)7-s + (−0.315 + 2.98i)9-s + (1.69 − 0.618i)11-s + (−1.30 + 1.09i)13-s + (−0.714 + 0.288i)15-s + (−1.80 + 1.03i)17-s + (2.98 + 1.72i)19-s + (1.88 + 3.02i)21-s + (−1.25 − 7.09i)23-s + (3.67 + 3.08i)25-s + (−4.20 + 3.05i)27-s + (−5.04 + 6.01i)29-s + (4.88 − 0.861i)31-s + ⋯
 L(s)  = 1 + (0.668 + 0.743i)3-s + (−0.0680 + 0.186i)5-s + (0.766 + 0.135i)7-s + (−0.105 + 0.994i)9-s + (0.512 − 0.186i)11-s + (−0.363 + 0.304i)13-s + (−0.184 + 0.0744i)15-s + (−0.436 + 0.252i)17-s + (0.685 + 0.395i)19-s + (0.412 + 0.659i)21-s + (−0.260 − 1.47i)23-s + (0.735 + 0.617i)25-s + (−0.809 + 0.587i)27-s + (−0.937 + 1.11i)29-s + (0.877 − 0.154i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$432$$    =    $$2^{4} \cdot 3^{3}$$ Sign: $0.459 - 0.887i$ Analytic conductor: $$3.44953$$ Root analytic conductor: $$1.85729$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{432} (335, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 432,\ (\ :1/2),\ 0.459 - 0.887i)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.54458 + 0.939440i$$ $$L(\frac12)$$ $$\approx$$ $$1.54458 + 0.939440i$$ $$L(\frac{3}{2})$$ 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 + (-1.15 - 1.28i)T$$
good5 $$1 + (0.152 - 0.417i)T + (-3.83 - 3.21i)T^{2}$$
7 $$1 + (-2.02 - 0.357i)T + (6.57 + 2.39i)T^{2}$$
11 $$1 + (-1.69 + 0.618i)T + (8.42 - 7.07i)T^{2}$$
13 $$1 + (1.30 - 1.09i)T + (2.25 - 12.8i)T^{2}$$
17 $$1 + (1.80 - 1.03i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (-2.98 - 1.72i)T + (9.5 + 16.4i)T^{2}$$
23 $$1 + (1.25 + 7.09i)T + (-21.6 + 7.86i)T^{2}$$
29 $$1 + (5.04 - 6.01i)T + (-5.03 - 28.5i)T^{2}$$
31 $$1 + (-4.88 + 0.861i)T + (29.1 - 10.6i)T^{2}$$
37 $$1 + (3.34 + 5.79i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-0.844 - 1.00i)T + (-7.11 + 40.3i)T^{2}$$
43 $$1 + (-0.952 - 2.61i)T + (-32.9 + 27.6i)T^{2}$$
47 $$1 + (-1.10 + 6.27i)T + (-44.1 - 16.0i)T^{2}$$
53 $$1 + 4.42iT - 53T^{2}$$
59 $$1 + (1.57 + 0.572i)T + (45.1 + 37.9i)T^{2}$$
61 $$1 + (-0.506 + 2.87i)T + (-57.3 - 20.8i)T^{2}$$
67 $$1 + (-6.09 - 7.25i)T + (-11.6 + 65.9i)T^{2}$$
71 $$1 + (4.39 + 7.60i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-7.57 + 13.1i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (-7.55 + 9.00i)T + (-13.7 - 77.7i)T^{2}$$
83 $$1 + (-3.93 - 3.29i)T + (14.4 + 81.7i)T^{2}$$
89 $$1 + (14.5 + 8.39i)T + (44.5 + 77.0i)T^{2}$$
97 $$1 + (-11.3 + 4.13i)T + (74.3 - 62.3i)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

−11.10781913861592828426625831418, −10.46882641957663603513756619162, −9.383649183163458419791214988961, −8.689062305566973425053260609919, −7.83805172113593999067385667185, −6.75133252562414359085888736742, −5.32377529500371240083628608911, −4.43087052915258518631638952175, −3.33113966018043814031837945708, −1.97243811338096047170925358205, 1.25565539261998633744893587302, 2.61561456222847727910875624369, 3.98128010152911881383117887028, 5.20126619905390985140648209846, 6.50109413285788881351196186798, 7.47210378246166908940072283466, 8.113107838414259331027365181866, 9.113508759466275323049297838242, 9.867629599389791325487881520338, 11.27315814427682182660435745615