Properties

Label 2-432-108.47-c1-0-0
Degree $2$
Conductor $432$
Sign $-0.950 + 0.311i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.950 + 0.311i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.950 + 0.311i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0340307 - 0.212828i\)
\(L(\frac12)\) \(\approx\) \(0.0340307 - 0.212828i\)
\(L(\frac{3}{2})\) 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

−11.78723698954545779033004503940, −10.59135135855683649417720904620, −9.899628096133394382892888113293, −9.305324445555330517919782194509, −7.79621630323198553072020967085, −7.16356942902432708599667362891, −5.76159707830715245753383234190, −4.82622829660452341381403049166, −4.01343349972055530047422254475, −2.50872624522362271946667089635, 0.13659284784905257028102931201, 2.02188415544648933406247451947, 3.56604201562958805664305271835, 5.01124229459051550595813837969, 5.97453979114853509370242609294, 6.87673800402805826403847399542, 7.993291070526100051506516996096, 8.429694462618049031968624877781, 9.947741488256875935040829027871, 10.81888283258336782830001592044

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