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 $0$

Origins

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Normalization:  

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

Graph of the $Z$-function along the critical line