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Properties

Label 2-432-27.7-c1-0-13
Degree $2$
Conductor $432$
Sign $-0.342 + 0.939i$
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  + (0.747 − 1.56i)3-s + (−0.738 − 4.19i)5-s + (2.50 + 2.10i)7-s + (−1.88 − 2.33i)9-s + (−0.0777 + 0.441i)11-s + (2.92 + 1.06i)13-s + (−7.09 − 1.97i)15-s + (−1.84 − 3.19i)17-s + (1.19 − 2.06i)19-s + (5.15 − 2.34i)21-s + (−5.22 + 4.38i)23-s + (−12.3 + 4.48i)25-s + (−5.05 + 1.19i)27-s + (0.616 − 0.224i)29-s + (3.54 − 2.97i)31-s + ⋯
L(s)  = 1  + (0.431 − 0.902i)3-s + (−0.330 − 1.87i)5-s + (0.946 + 0.793i)7-s + (−0.627 − 0.778i)9-s + (−0.0234 + 0.133i)11-s + (0.810 + 0.294i)13-s + (−1.83 − 0.510i)15-s + (−0.446 − 0.773i)17-s + (0.273 − 0.474i)19-s + (1.12 − 0.510i)21-s + (−1.09 + 0.915i)23-s + (−2.46 + 0.896i)25-s + (−0.973 + 0.229i)27-s + (0.114 − 0.0416i)29-s + (0.635 − 0.533i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.342 + 0.939i$
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: \((2,\ 432,\ (\ :1/2),\ -0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.901950 - 1.28906i\)
\(L(\frac12)\) \(\approx\) \(0.901950 - 1.28906i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.747 + 1.56i)T \)
good5 \( 1 + (0.738 + 4.19i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-2.50 - 2.10i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.0777 - 0.441i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (-2.92 - 1.06i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (1.84 + 3.19i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.19 + 2.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (5.22 - 4.38i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-0.616 + 0.224i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-3.54 + 2.97i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (0.459 + 0.795i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.96 - 1.07i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.522 - 2.96i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-6.99 - 5.86i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 10.8T + 53T^{2} \)
59 \( 1 + (0.976 + 5.53i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-9.68 - 8.12i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (8.22 + 2.99i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-4.81 - 8.34i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-7.57 + 13.1i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.69 - 1.34i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-9.79 + 3.56i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (3.11 - 5.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.477 - 2.70i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.46888857719727189048269879288, −9.480763101154077918571348069893, −8.863582684430929642426520760023, −8.239830206147533951867193419719, −7.52194074480277010663419311854, −5.99523556513424606781225545922, −5.12669810378636229329526721301, −4.06048773269742590171606237441, −2.18461180167000643231043752136, −1.04208476394416494740878992334, 2.33751715710662693437059353478, 3.60974492979638586630640540684, 4.19878934337729461092280685712, 5.78333794708771999313118926164, 6.86022218182985274584234563760, 7.900416439348378364380872285882, 8.489493837722340608998657297032, 10.08017959732631153286902906527, 10.57573729194989583409712755863, 10.99981048992679474356344108493

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