Properties

Label 2-432-108.23-c1-0-3
Degree $2$
Conductor $432$
Sign $-0.205 - 0.978i$
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  + (0.939 + 1.45i)3-s + (−1.07 − 0.188i)5-s + (0.0466 + 0.0555i)7-s + (−1.23 + 2.73i)9-s + (0.889 + 5.04i)11-s + (−5.31 + 1.93i)13-s + (−0.730 − 1.73i)15-s + (3.79 + 2.19i)17-s + (4.96 − 2.86i)19-s + (−0.0370 + 0.119i)21-s + (3.14 + 2.64i)23-s + (−3.58 − 1.30i)25-s + (−5.13 + 0.767i)27-s + (−1.30 + 3.58i)29-s + (2.61 − 3.11i)31-s + ⋯
L(s)  = 1  + (0.542 + 0.840i)3-s + (−0.478 − 0.0844i)5-s + (0.0176 + 0.0209i)7-s + (−0.412 + 0.911i)9-s + (0.268 + 1.52i)11-s + (−1.47 + 0.536i)13-s + (−0.188 − 0.448i)15-s + (0.920 + 0.531i)17-s + (1.13 − 0.657i)19-s + (−0.00808 + 0.0261i)21-s + (0.656 + 0.550i)23-s + (−0.717 − 0.261i)25-s + (−0.989 + 0.147i)27-s + (−0.242 + 0.665i)29-s + (0.468 − 0.558i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.205 - 0.978i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.205 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.863405 + 1.06300i\)
\(L(\frac12)\) \(\approx\) \(0.863405 + 1.06300i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 - 1.45i)T \)
good5 \( 1 + (1.07 + 0.188i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.0466 - 0.0555i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.889 - 5.04i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (5.31 - 1.93i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-3.79 - 2.19i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.96 + 2.86i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.14 - 2.64i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (1.30 - 3.58i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.61 + 3.11i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (1.14 - 1.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.494 + 1.35i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.128 - 0.0227i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (-4.26 + 3.57i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 10.4iT - 53T^{2} \)
59 \( 1 + (1.69 - 9.59i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-4.96 + 4.16i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-2.28 - 6.27i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.47 - 6.02i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.77 - 4.80i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.83 + 13.2i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (-3.77 - 1.37i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (-14.4 + 8.34i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.74 + 15.5i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.55351710979233352936714152471, −10.08481899400527943680139727605, −9.795512121529721313510848833455, −8.861448196803716324535133056320, −7.63310300810646173192811333173, −7.14023635171111174883312190133, −5.30059758249189609675215000584, −4.58578601465351925689244605029, −3.52000964171218268320953018528, −2.16056295332672546349227273782, 0.844704991097303548589962971227, 2.75267961710248200511759343163, 3.56031787658944474676081527844, 5.26196527494168848197109840579, 6.26146611443148349509989536649, 7.59061877702820495658734283566, 7.80082077177925053049768005634, 9.001534493838347752327262699024, 9.842477928456056028936699033254, 11.08619410105186812129480553188

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