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Properties

Label 2-432-108.59-c1-0-3
Degree $2$
Conductor $432$
Sign $0.0320 - 0.999i$
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  + (−1.35 + 1.07i)3-s + (1.16 + 3.20i)5-s + (4.32 − 0.763i)7-s + (0.697 − 2.91i)9-s + (−3.88 − 1.41i)11-s + (4.63 + 3.89i)13-s + (−5.01 − 3.10i)15-s + (0.945 + 0.546i)17-s + (−2.45 + 1.41i)19-s + (−5.06 + 5.68i)21-s + (−0.327 + 1.85i)23-s + (−5.05 + 4.24i)25-s + (2.18 + 4.71i)27-s + (−3.45 − 4.12i)29-s + (1.52 + 0.268i)31-s + ⋯
L(s)  = 1  + (−0.785 + 0.619i)3-s + (0.520 + 1.43i)5-s + (1.63 − 0.288i)7-s + (0.232 − 0.972i)9-s + (−1.17 − 0.426i)11-s + (1.28 + 1.07i)13-s + (−1.29 − 0.800i)15-s + (0.229 + 0.132i)17-s + (−0.562 + 0.324i)19-s + (−1.10 + 1.23i)21-s + (−0.0681 + 0.386i)23-s + (−1.01 + 0.848i)25-s + (0.419 + 0.907i)27-s + (−0.642 − 0.765i)29-s + (0.273 + 0.0481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0320 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0320 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.0320 - 0.999i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.0320 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.950051 + 0.920046i\)
\(L(\frac12)\) \(\approx\) \(0.950051 + 0.920046i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 - 1.07i)T \)
good5 \( 1 + (-1.16 - 3.20i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (-4.32 + 0.763i)T + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (3.88 + 1.41i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (-4.63 - 3.89i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.945 - 0.546i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.45 - 1.41i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.327 - 1.85i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.45 + 4.12i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-1.52 - 0.268i)T + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (2.48 - 4.31i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.29 - 3.92i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-1.43 + 3.93i)T + (-32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.51 - 8.60i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 12.1iT - 53T^{2} \)
59 \( 1 + (-3.09 + 1.12i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (0.463 + 2.62i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-2.32 + 2.77i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (0.842 - 1.45i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.58 + 9.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.26 + 7.46i)T + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (0.240 - 0.201i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (-6.60 + 3.81i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.903 - 0.328i)T + (74.3 + 62.3i)T^{2} \)
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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

−11.19865323252259208338431997802, −10.66377360589182357361499562172, −9.986579591058298464291707763075, −8.630180156002240135895018968313, −7.63600713937917125358416638077, −6.49018497057185018427215324750, −5.74232587077289042921877401201, −4.64213816700328152262790803455, −3.48997070973467796850057178177, −1.83626086068116857213346047672, 1.03363767618060245263022119866, 2.10883778508925389218447896740, 4.53339203075232413372264827665, 5.33883414367633301461301269350, 5.73116194035680112590758637016, 7.40654413467655757394075608466, 8.277265774517082126392120522489, 8.745321167099796961938544891520, 10.35644923614826823924842181850, 10.93414443316968364132223619279

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