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Properties

Label 2-432-27.7-c1-0-2
Degree $2$
Conductor $432$
Sign $0.513 - 0.858i$
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  + (−0.887 − 1.48i)3-s + (0.444 + 2.51i)5-s + (0.612 + 0.513i)7-s + (−1.42 + 2.63i)9-s + (−0.313 + 1.77i)11-s + (−2.44 − 0.888i)13-s + (3.35 − 2.89i)15-s + (3.13 + 5.43i)17-s + (−2.88 + 5.00i)19-s + (0.220 − 1.36i)21-s + (−1.80 + 1.51i)23-s + (−1.45 + 0.528i)25-s + (5.19 − 0.222i)27-s + (7.05 − 2.56i)29-s + (4.55 − 3.81i)31-s + ⋯
L(s)  = 1  + (−0.512 − 0.858i)3-s + (0.198 + 1.12i)5-s + (0.231 + 0.194i)7-s + (−0.475 + 0.879i)9-s + (−0.0946 + 0.536i)11-s + (−0.677 − 0.246i)13-s + (0.865 − 0.747i)15-s + (0.760 + 1.31i)17-s + (−0.662 + 1.14i)19-s + (0.0481 − 0.298i)21-s + (−0.377 + 0.316i)23-s + (−0.290 + 0.105i)25-s + (0.999 − 0.0428i)27-s + (1.31 − 0.476i)29-s + (0.817 − 0.685i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.513 - 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.513 - 0.858i$
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: \((2,\ 432,\ (\ :1/2),\ 0.513 - 0.858i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.917037 + 0.520131i\)
\(L(\frac12)\) \(\approx\) \(0.917037 + 0.520131i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.887 + 1.48i)T \)
good5 \( 1 + (-0.444 - 2.51i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-0.612 - 0.513i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.313 - 1.77i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.44 + 0.888i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.88 - 5.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.80 - 1.51i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.05 + 2.56i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-4.55 + 3.81i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-0.0710 - 0.123i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.49 - 2.72i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (2.02 - 11.4i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-2.93 - 2.46i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + (0.688 + 3.90i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.0 + 8.39i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (13.7 + 5.01i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-3.27 - 5.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.483 - 0.837i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.693 + 0.252i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (-6.18 + 2.25i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.90 - 5.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.957 - 5.43i)T + (-91.1 - 33.1i)T^{2} \)
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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

−11.29900740625858803940598129481, −10.44630532367766766994948188037, −9.849052807548470100252673273307, −8.077891666292537760639566015701, −7.72906528254185996646808497002, −6.39861653597530506188108739391, −6.03868608062283530249469810259, −4.59888337212364641629061476048, −2.93741327430357324466801374754, −1.77881888491871320625664133316, 0.73392344825847963496727386482, 2.92617490651327774122110299830, 4.57385277589465322684413731681, 4.91598191048454571011808386602, 6.03319951535323349345293665948, 7.27256275914191025041985928904, 8.654342729359469846488556320340, 9.138804371648405015150330829620, 10.12616457112992204565166718131, 10.92007988136387399634684493082

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