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Properties

Label 2-432-108.83-c1-0-1
Degree $2$
Conductor $432$
Sign $0.642 - 0.766i$
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  + (−1.16 + 1.28i)3-s + (−1.92 − 2.29i)5-s + (−0.0588 − 0.161i)7-s + (−0.290 − 2.98i)9-s + (4.37 + 3.67i)11-s + (0.936 + 5.31i)13-s + (5.17 + 0.200i)15-s + (2.85 − 1.64i)17-s + (5.19 + 3.00i)19-s + (0.275 + 0.112i)21-s + (−4.55 − 1.65i)23-s + (−0.685 + 3.88i)25-s + (4.16 + 3.10i)27-s + (−0.654 − 0.115i)29-s + (−2.74 + 7.54i)31-s + ⋯
L(s)  = 1  + (−0.671 + 0.740i)3-s + (−0.859 − 1.02i)5-s + (−0.0222 − 0.0611i)7-s + (−0.0968 − 0.995i)9-s + (1.31 + 1.10i)11-s + (0.259 + 1.47i)13-s + (1.33 + 0.0518i)15-s + (0.692 − 0.400i)17-s + (1.19 + 0.688i)19-s + (0.0602 + 0.0246i)21-s + (−0.949 − 0.345i)23-s + (−0.137 + 0.777i)25-s + (0.802 + 0.597i)27-s + (−0.121 − 0.0214i)29-s + (−0.493 + 1.35i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.898456 + 0.418981i\)
\(L(\frac12)\) \(\approx\) \(0.898456 + 0.418981i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.16 - 1.28i)T \)
good5 \( 1 + (1.92 + 2.29i)T + (-0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.0588 + 0.161i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (-4.37 - 3.67i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-0.936 - 5.31i)T + (-12.2 + 4.44i)T^{2} \)
17 \( 1 + (-2.85 + 1.64i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.19 - 3.00i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.55 + 1.65i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (0.654 + 0.115i)T + (27.2 + 9.91i)T^{2} \)
31 \( 1 + (2.74 - 7.54i)T + (-23.7 - 19.9i)T^{2} \)
37 \( 1 + (-2.25 - 3.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.59 + 1.51i)T + (38.5 - 14.0i)T^{2} \)
43 \( 1 + (-1.57 + 1.87i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-5.36 + 1.95i)T + (36.0 - 30.2i)T^{2} \)
53 \( 1 + 5.08iT - 53T^{2} \)
59 \( 1 + (3.98 - 3.34i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-3.78 + 1.37i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (14.1 - 2.50i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-1.47 - 2.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (3.06 - 5.30i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-9.23 - 1.62i)T + (74.2 + 27.0i)T^{2} \)
83 \( 1 + (-0.641 + 3.63i)T + (-77.9 - 28.3i)T^{2} \)
89 \( 1 + (0.248 + 0.143i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-9.32 - 7.82i)T + (16.8 + 95.5i)T^{2} \)
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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

−11.75103406506361528991217277136, −10.29447042743790943507879694422, −9.383137005837371210436527704912, −8.894421958909720536834087180370, −7.53417566953354758494956777981, −6.57287745872574147214724986192, −5.32647020337370557457036058044, −4.30369796208126125441116199252, −3.84278348744451383122785556633, −1.28547976276075373937894769843, 0.852917880443753682943194532580, 2.92400579255607780675594959283, 3.91150943919596632609993515409, 5.71729647670707713057618269137, 6.18566282614418888114263055510, 7.58628679122439019575101791652, 7.73651156007429294724457035806, 9.187779493700050257297734496671, 10.52240296349835651226189475218, 11.18911464070045485668218507896

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