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Properties

Label 2-432-27.7-c1-0-16
Degree $2$
Conductor $432$
Sign $-0.860 - 0.509i$
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  + (−1.51 − 0.842i)3-s + (−0.672 − 3.81i)5-s + (−2.31 − 1.94i)7-s + (1.58 + 2.54i)9-s + (−0.624 + 3.54i)11-s + (−3.29 − 1.20i)13-s + (−2.19 + 6.33i)15-s + (3.25 + 5.62i)17-s + (1.15 − 1.99i)19-s + (1.86 + 4.88i)21-s + (0.461 − 0.387i)23-s + (−9.38 + 3.41i)25-s + (−0.244 − 5.19i)27-s + (−3.20 + 1.16i)29-s + (−2.33 + 1.95i)31-s + ⋯
L(s)  = 1  + (−0.873 − 0.486i)3-s + (−0.300 − 1.70i)5-s + (−0.873 − 0.733i)7-s + (0.526 + 0.849i)9-s + (−0.188 + 1.06i)11-s + (−0.914 − 0.333i)13-s + (−0.566 + 1.63i)15-s + (0.788 + 1.36i)17-s + (0.264 − 0.458i)19-s + (0.406 + 1.06i)21-s + (0.0962 − 0.0807i)23-s + (−1.87 + 0.683i)25-s + (−0.0469 − 0.998i)27-s + (−0.595 + 0.216i)29-s + (−0.418 + 0.351i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.860 - 0.509i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (385, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.860 - 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0770577 + 0.281282i\)
\(L(\frac12)\) \(\approx\) \(0.0770577 + 0.281282i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 + 0.842i)T \)
good5 \( 1 + (0.672 + 3.81i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.31 + 1.94i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.624 - 3.54i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (3.29 + 1.20i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.25 - 5.62i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.15 + 1.99i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.461 + 0.387i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (3.20 - 1.16i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (2.33 - 1.95i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (2.29 + 3.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (9.50 + 3.45i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.70 + 9.67i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-1.11 - 0.937i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 5.64T + 53T^{2} \)
59 \( 1 + (-1.03 - 5.85i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-4.48 - 3.76i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (6.06 + 2.20i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (0.397 + 0.689i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.747 - 1.29i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.2 - 3.73i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-10.5 + 3.85i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-4.55 + 7.88i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.96 + 11.1i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.47365331874635724936803326677, −9.907047979774459798667848816739, −8.798773676798617871792862469935, −7.64622387907326798336386085378, −7.04137056809577492732998027090, −5.64645405640886827785471756915, −4.92495156244137317208386446829, −3.90532463914548983167465694878, −1.65157120233554522907329012901, −0.20286171198837415169163170397, 2.84909896657383977828600414121, 3.50430150590280992700227357372, 5.17197509963762761012384102240, 6.12580841239098905589366391722, 6.81340422543452227352533522400, 7.76844283150088705361517074113, 9.437157237209544032941709718501, 9.903181598084967534261683202539, 10.86328593918840468950586789765, 11.61677482639501637664824392885

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