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Properties

Label 2-432-108.11-c1-0-12
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: $\chi_{432} (335, \cdot )$
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

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

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