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Properties

Label 2-432-108.23-c1-0-6
Degree $2$
Conductor $432$
Sign $0.937 + 0.349i$
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  + (−1.36 − 1.06i)3-s + (1.83 + 0.322i)5-s + (0.441 + 0.526i)7-s + (0.733 + 2.90i)9-s + (0.0632 + 0.358i)11-s + (2.50 − 0.913i)13-s + (−2.15 − 2.38i)15-s + (2.50 + 1.44i)17-s + (2.50 − 1.44i)19-s + (−0.0430 − 1.18i)21-s + (1.39 + 1.16i)23-s + (−1.45 − 0.528i)25-s + (2.09 − 4.75i)27-s + (2.03 − 5.59i)29-s + (2.94 − 3.50i)31-s + ⋯
L(s)  = 1  + (−0.788 − 0.614i)3-s + (0.818 + 0.144i)5-s + (0.166 + 0.198i)7-s + (0.244 + 0.969i)9-s + (0.0190 + 0.108i)11-s + (0.696 − 0.253i)13-s + (−0.556 − 0.616i)15-s + (0.606 + 0.350i)17-s + (0.575 − 0.332i)19-s + (−0.00940 − 0.259i)21-s + (0.289 + 0.243i)23-s + (−0.290 − 0.105i)25-s + (0.403 − 0.915i)27-s + (0.378 − 1.03i)29-s + (0.528 − 0.630i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.937 + 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.937 + 0.349i$
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: \((2,\ 432,\ (\ :1/2),\ 0.937 + 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29948 - 0.234285i\)
\(L(\frac12)\) \(\approx\) \(1.29948 - 0.234285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.36 + 1.06i)T \)
good5 \( 1 + (-1.83 - 0.322i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.441 - 0.526i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.0632 - 0.358i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-2.50 + 0.913i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.50 - 1.44i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.50 + 1.44i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.39 - 1.16i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-2.03 + 5.59i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (-2.94 + 3.50i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (-1.77 + 3.07i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-3.29 - 9.05i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-11.8 + 2.08i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (7.61 - 6.39i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + 1.01iT - 53T^{2} \)
59 \( 1 + (0.864 - 4.90i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (9.61 - 8.06i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (1.71 + 4.70i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (8.29 - 14.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.84 + 11.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.52 - 6.93i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (3.65 + 1.33i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (12.1 - 6.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.204 + 1.16i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.20313030032592307507655178483, −10.27374936014242572750016655255, −9.477657819964550972335042143660, −8.188011863404233271359674996399, −7.35090805076356895603119837121, −6.07655340981831580702955491286, −5.77430256862313203459924526794, −4.44235820643235370735823647479, −2.64437059829174547622512865765, −1.25932958996926999375101204634, 1.28699232069859609563574882304, 3.25819975900031276084981614486, 4.53737401460980584405508857604, 5.52235861182284630198655730780, 6.23845743448762071473123469978, 7.36287936944614126469926624319, 8.756843374857156921575408474787, 9.541122474034005193802643717605, 10.34938834609750485125953544845, 11.06645387872241821175096801201

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