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Properties

Label 2-432-16.5-c1-0-7
Degree $2$
Conductor $432$
Sign $-0.636 - 0.771i$
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  + (0.636 + 1.26i)2-s + (−1.18 + 1.60i)4-s + (1.27 + 1.27i)5-s − 0.0285i·7-s + (−2.78 − 0.477i)8-s + (−0.798 + 2.42i)10-s + (2.94 + 2.94i)11-s + (−1.34 + 1.34i)13-s + (0.0360 − 0.0181i)14-s + (−1.17 − 3.82i)16-s + 1.40·17-s + (−5.46 + 5.46i)19-s + (−3.56 + 0.534i)20-s + (−1.84 + 5.58i)22-s − 0.0963i·23-s + ⋯
L(s)  = 1  + (0.450 + 0.892i)2-s + (−0.594 + 0.804i)4-s + (0.570 + 0.570i)5-s − 0.0107i·7-s + (−0.985 − 0.168i)8-s + (−0.252 + 0.766i)10-s + (0.886 + 0.886i)11-s + (−0.373 + 0.373i)13-s + (0.00963 − 0.00485i)14-s + (−0.292 − 0.956i)16-s + 0.339·17-s + (−1.25 + 1.25i)19-s + (−0.797 + 0.119i)20-s + (−0.392 + 1.19i)22-s − 0.0200i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $-0.636 - 0.771i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (325, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ -0.636 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.724562 + 1.53749i\)
\(L(\frac12)\) \(\approx\) \(0.724562 + 1.53749i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.636 - 1.26i)T \)
3 \( 1 \)
good5 \( 1 + (-1.27 - 1.27i)T + 5iT^{2} \)
7 \( 1 + 0.0285iT - 7T^{2} \)
11 \( 1 + (-2.94 - 2.94i)T + 11iT^{2} \)
13 \( 1 + (1.34 - 1.34i)T - 13iT^{2} \)
17 \( 1 - 1.40T + 17T^{2} \)
19 \( 1 + (5.46 - 5.46i)T - 19iT^{2} \)
23 \( 1 + 0.0963iT - 23T^{2} \)
29 \( 1 + (-6.62 + 6.62i)T - 29iT^{2} \)
31 \( 1 + 3.75T + 31T^{2} \)
37 \( 1 + (-4.36 - 4.36i)T + 37iT^{2} \)
41 \( 1 - 5.24iT - 41T^{2} \)
43 \( 1 + (5.87 + 5.87i)T + 43iT^{2} \)
47 \( 1 - 3.04T + 47T^{2} \)
53 \( 1 + (-0.358 - 0.358i)T + 53iT^{2} \)
59 \( 1 + (-4.94 - 4.94i)T + 59iT^{2} \)
61 \( 1 + (-6.84 + 6.84i)T - 61iT^{2} \)
67 \( 1 + (-1.56 + 1.56i)T - 67iT^{2} \)
71 \( 1 + 13.1iT - 71T^{2} \)
73 \( 1 + 7.44iT - 73T^{2} \)
79 \( 1 + 2.58T + 79T^{2} \)
83 \( 1 + (-7.92 + 7.92i)T - 83iT^{2} \)
89 \( 1 - 9.64iT - 89T^{2} \)
97 \( 1 - 2.24T + 97T^{2} \)
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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

−11.83365405424818572256729843045, −10.31395723479461906139525183860, −9.674348763385216554852847689628, −8.597296280660320705680320366788, −7.62456713605882369378135115597, −6.56875552003523380120815640253, −6.12558366737527100296284407360, −4.73294379544239812753681895727, −3.81822491982475630269882296577, −2.25126556678567162608621874025, 1.01268470998944058569113321529, 2.50087598919314477085092979060, 3.78607155214651405170336817184, 4.93779359381350614125542644829, 5.78623020041615127678338881152, 6.84628446786406897957684431457, 8.621255234600999555964393058854, 9.035312679146809006858871954330, 10.06232867973589153034463250677, 10.93594041595162882808524042446

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