Properties

Label 2-432-108.11-c1-0-6
Degree $2$
Conductor $432$
Sign $0.999 - 0.00766i$
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  + (−1.48 + 0.898i)3-s + (0.357 − 0.982i)5-s + (0.862 + 0.152i)7-s + (1.38 − 2.66i)9-s + (−0.855 + 0.311i)11-s + (0.662 − 0.555i)13-s + (0.353 + 1.77i)15-s + (4.52 − 2.61i)17-s + (3.63 + 2.09i)19-s + (−1.41 + 0.550i)21-s + (−0.356 − 2.02i)23-s + (2.99 + 2.51i)25-s + (0.341 + 5.18i)27-s + (0.526 − 0.627i)29-s + (8.52 − 1.50i)31-s + ⋯
L(s)  = 1  + (−0.854 + 0.518i)3-s + (0.159 − 0.439i)5-s + (0.326 + 0.0575i)7-s + (0.461 − 0.887i)9-s + (−0.257 + 0.0938i)11-s + (0.183 − 0.154i)13-s + (0.0912 + 0.458i)15-s + (1.09 − 0.633i)17-s + (0.834 + 0.481i)19-s + (−0.308 + 0.120i)21-s + (−0.0743 − 0.421i)23-s + (0.598 + 0.502i)25-s + (0.0657 + 0.997i)27-s + (0.0977 − 0.116i)29-s + (1.53 − 0.269i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.00766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.999 - 0.00766i$
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: \((2,\ 432,\ (\ :1/2),\ 0.999 - 0.00766i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16282 + 0.00445881i\)
\(L(\frac12)\) \(\approx\) \(1.16282 + 0.00445881i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.48 - 0.898i)T \)
good5 \( 1 + (-0.357 + 0.982i)T + (-3.83 - 3.21i)T^{2} \)
7 \( 1 + (-0.862 - 0.152i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (0.855 - 0.311i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-0.662 + 0.555i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-4.52 + 2.61i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.63 - 2.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.356 + 2.02i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-0.526 + 0.627i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-8.52 + 1.50i)T + (29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.44 - 2.49i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.04 - 6.00i)T + (-7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.0722 + 0.198i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (-1.30 + 7.38i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + 8.58iT - 53T^{2} \)
59 \( 1 + (11.8 + 4.29i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.82 - 10.3i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-1.37 - 1.63i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.76 + 3.06i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.656 - 1.13i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.814 + 0.970i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (9.32 + 7.82i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-4.27 - 2.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (13.6 - 4.95i)T + (74.3 - 62.3i)T^{2} \)
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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

−11.24871498342065039707483561981, −10.13523668787394861766541840717, −9.657708245376956399685030548412, −8.470951374343649779727692119833, −7.44582026513677160303273777122, −6.22428663993248955358208795390, −5.30898003607939890176247631589, −4.58701077653980968313653486547, −3.17719122742178025301244798724, −1.08782610840144141621294478234, 1.24391346617201218073583105343, 2.87054677160541241992766279726, 4.49980076520728393385753502776, 5.58005559921625378235713997472, 6.38468037509057893514556038171, 7.41673048897383049617155709409, 8.138145934635975834788744060516, 9.528141802817980089630858384989, 10.52206846979148581323376202366, 11.07892120339953751005632500952

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