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Properties

Label 2-432-27.7-c1-0-10
Degree $2$
Conductor $432$
Sign $0.914 - 0.403i$
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  + (1.70 + 0.329i)3-s + (0.198 + 1.12i)5-s + (−0.914 − 0.767i)7-s + (2.78 + 1.11i)9-s + (0.411 − 2.33i)11-s + (4.28 + 1.55i)13-s + (−0.0328 + 1.97i)15-s + (2.15 + 3.72i)17-s + (0.315 − 0.547i)19-s + (−1.30 − 1.60i)21-s + (−1.05 + 0.887i)23-s + (3.47 − 1.26i)25-s + (4.36 + 2.81i)27-s + (−9.61 + 3.50i)29-s + (−3.28 + 2.75i)31-s + ⋯
L(s)  = 1  + (0.981 + 0.189i)3-s + (0.0885 + 0.502i)5-s + (−0.345 − 0.289i)7-s + (0.927 + 0.373i)9-s + (0.124 − 0.704i)11-s + (1.18 + 0.432i)13-s + (−0.00848 + 0.509i)15-s + (0.522 + 0.904i)17-s + (0.0724 − 0.125i)19-s + (−0.284 − 0.350i)21-s + (−0.220 + 0.185i)23-s + (0.695 − 0.253i)25-s + (0.840 + 0.542i)27-s + (−1.78 + 0.650i)29-s + (−0.589 + 0.494i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.914 - 0.403i$
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: \((2,\ 432,\ (\ :1/2),\ 0.914 - 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93020 + 0.406905i\)
\(L(\frac12)\) \(\approx\) \(1.93020 + 0.406905i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.329i)T \)
good5 \( 1 + (-0.198 - 1.12i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.914 + 0.767i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.411 + 2.33i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-4.28 - 1.55i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.15 - 3.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.315 + 0.547i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.05 - 0.887i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (9.61 - 3.50i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (3.28 - 2.75i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (4.23 + 7.33i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.42 + 1.24i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.34 + 7.62i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (1.89 + 1.58i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 1.65T + 53T^{2} \)
59 \( 1 + (-2.05 - 11.6i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-1.11 - 0.932i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (14.0 + 5.11i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.74 + 6.48i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.22 + 9.05i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-13.0 + 4.75i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.64 - 1.69i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-4.93 + 8.55i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.74 - 9.90i)T + (-91.1 - 33.1i)T^{2} \)
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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

−10.78730935030453753796407391981, −10.51530717541579310027183267209, −9.174390269800941071968304488412, −8.687561731356695108874386291680, −7.56403019898430928576084938820, −6.68280578347061337340496294280, −5.55588127569049556295580175998, −3.83901065389851874903056362101, −3.37661009884730658132342027328, −1.75347727429245244162801493426, 1.47123579559413192475271130030, 2.93567953242465541645253023387, 4.00614753896276069261137843609, 5.29942669173232757387619378665, 6.52045682416205010913761825380, 7.58026026726256817825050875918, 8.389462529255364391743712334486, 9.329672820058544189993510987456, 9.814468496814389043542285579661, 11.10097860820505215567900209064

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