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Properties

Label 2-432-16.13-c1-0-17
Degree $2$
Conductor $432$
Sign $0.569 - 0.822i$
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  + (1.27 + 0.610i)2-s + (1.25 + 1.55i)4-s + (0.486 − 0.486i)5-s + 0.822i·7-s + (0.652 + 2.75i)8-s + (0.917 − 0.323i)10-s + (0.518 − 0.518i)11-s + (2.51 + 2.51i)13-s + (−0.501 + 1.04i)14-s + (−0.846 + 3.90i)16-s + 1.21·17-s + (−0.571 − 0.571i)19-s + (1.36 + 0.146i)20-s + (0.977 − 0.344i)22-s − 6.93i·23-s + ⋯
L(s)  = 1  + (0.902 + 0.431i)2-s + (0.627 + 0.778i)4-s + (0.217 − 0.217i)5-s + 0.310i·7-s + (0.230 + 0.973i)8-s + (0.290 − 0.102i)10-s + (0.156 − 0.156i)11-s + (0.697 + 0.697i)13-s + (−0.134 + 0.280i)14-s + (−0.211 + 0.977i)16-s + 0.293·17-s + (−0.131 − 0.131i)19-s + (0.305 + 0.0327i)20-s + (0.208 − 0.0735i)22-s − 1.44i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.569 - 0.822i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(432\)    =    \(2^{4} \cdot 3^{3}\)
Sign: $0.569 - 0.822i$
Analytic conductor: \(3.44953\)
Root analytic conductor: \(1.85729\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{432} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 432,\ (\ :1/2),\ 0.569 - 0.822i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.16844 + 1.13571i\)
\(L(\frac12)\) \(\approx\) \(2.16844 + 1.13571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.610i)T \)
3 \( 1 \)
good5 \( 1 + (-0.486 + 0.486i)T - 5iT^{2} \)
7 \( 1 - 0.822iT - 7T^{2} \)
11 \( 1 + (-0.518 + 0.518i)T - 11iT^{2} \)
13 \( 1 + (-2.51 - 2.51i)T + 13iT^{2} \)
17 \( 1 - 1.21T + 17T^{2} \)
19 \( 1 + (0.571 + 0.571i)T + 19iT^{2} \)
23 \( 1 + 6.93iT - 23T^{2} \)
29 \( 1 + (2.73 + 2.73i)T + 29iT^{2} \)
31 \( 1 + 3.84T + 31T^{2} \)
37 \( 1 + (-3.38 + 3.38i)T - 37iT^{2} \)
41 \( 1 + 11.0iT - 41T^{2} \)
43 \( 1 + (2.46 - 2.46i)T - 43iT^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 + (4.98 - 4.98i)T - 53iT^{2} \)
59 \( 1 + (-4.59 + 4.59i)T - 59iT^{2} \)
61 \( 1 + (-5.85 - 5.85i)T + 61iT^{2} \)
67 \( 1 + (9.63 + 9.63i)T + 67iT^{2} \)
71 \( 1 + 6.26iT - 71T^{2} \)
73 \( 1 + 8.86iT - 73T^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (-9.54 - 9.54i)T + 83iT^{2} \)
89 \( 1 + 10.8iT - 89T^{2} \)
97 \( 1 - 5.88T + 97T^{2} \)
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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

−11.44376775380852359185894189996, −10.68677897690691882437210918009, −9.254976113925016517253521699373, −8.511198329503688796021374960516, −7.40844052157887564106015110914, −6.40735130716356269385763805169, −5.62494052716461639972291349364, −4.52647454947945522380270458445, −3.47302044463127345857910835237, −2.03138585529163763168236658711, 1.46791928895586353248764087145, 3.02061725472155929606290616167, 3.96954364527647421002796538745, 5.22309712844338888259940081849, 6.09418618544956398299786098708, 7.06575238142151539492151007109, 8.164934574524772950117784639912, 9.589144052467223290791909135820, 10.25761063319402838852798717713, 11.17182515311708646468140682466

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