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Properties

Label 2-432-108.23-c1-0-10
Degree $2$
Conductor $432$
Sign $0.824 + 0.566i$
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  + (1.65 − 0.517i)3-s + (−2.52 − 0.445i)5-s + (1.40 + 1.67i)7-s + (2.46 − 1.71i)9-s + (−0.751 − 4.26i)11-s + (5.33 − 1.94i)13-s + (−4.41 + 0.572i)15-s + (5.53 + 3.19i)17-s + (2.85 − 1.64i)19-s + (3.19 + 2.04i)21-s + (−3.31 − 2.78i)23-s + (1.49 + 0.545i)25-s + (3.18 − 4.10i)27-s + (0.131 − 0.362i)29-s + (−4.37 + 5.21i)31-s + ⋯
L(s)  = 1  + (0.954 − 0.298i)3-s + (−1.13 − 0.199i)5-s + (0.532 + 0.634i)7-s + (0.821 − 0.570i)9-s + (−0.226 − 1.28i)11-s + (1.47 − 0.538i)13-s + (−1.13 + 0.147i)15-s + (1.34 + 0.775i)17-s + (0.654 − 0.378i)19-s + (0.697 + 0.446i)21-s + (−0.691 − 0.580i)23-s + (0.299 + 0.109i)25-s + (0.613 − 0.789i)27-s + (0.0245 − 0.0673i)29-s + (−0.785 + 0.935i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.824 + 0.566i$
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: \((2,\ 432,\ (\ :1/2),\ 0.824 + 0.566i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.68705 - 0.523542i\)
\(L(\frac12)\) \(\approx\) \(1.68705 - 0.523542i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.65 + 0.517i)T \)
good5 \( 1 + (2.52 + 0.445i)T + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.40 - 1.67i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.751 + 4.26i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (-5.33 + 1.94i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-5.53 - 3.19i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.85 + 1.64i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.31 + 2.78i)T + (3.99 + 22.6i)T^{2} \)
29 \( 1 + (-0.131 + 0.362i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.37 - 5.21i)T + (-5.38 - 30.5i)T^{2} \)
37 \( 1 + (3.81 - 6.61i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.138 + 0.380i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (9.35 - 1.64i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (6.74 - 5.65i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 - 7.00iT - 53T^{2} \)
59 \( 1 + (0.296 - 1.68i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-8.66 + 7.26i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-0.683 - 1.87i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-1.85 + 3.21i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-5.37 - 9.31i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.34 + 6.42i)T + (-60.5 - 50.7i)T^{2} \)
83 \( 1 + (1.31 + 0.479i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (6.85 - 3.95i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-2.09 - 11.8i)T + (-91.1 + 33.1i)T^{2} \)
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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

−11.20453820749951211318885122074, −10.18330553858597210280653740321, −8.797659298960346412201132827549, −8.220716732810443187392806272746, −7.918972840623806277253021711392, −6.43264365588052050009757919718, −5.30039477768763743443051896345, −3.73400628478469041279065715083, −3.18267482322682430884824684866, −1.26817040130755446162177919607, 1.70201540289951393973345379569, 3.53170078974778147466644932635, 3.99709587458684888691872395251, 5.22213816316608004371611425851, 7.06770374006734391010573221263, 7.65870238856113995156205202916, 8.280578968983002741727012761219, 9.500862581595203930738582632075, 10.21298371369811861012846019504, 11.28443712701730952239845053588

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