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Properties

Label 2-6e3-216.139-c2-0-53
Degree $2$
Conductor $216$
Sign $0.979 - 0.203i$
Analytic cond. $5.88557$
Root an. cond. $2.42602$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.87 + 0.684i)2-s + (2.61 − 1.47i)3-s + (3.06 + 2.57i)4-s + (5.91 − 0.987i)6-s + (4.00 + 6.92i)8-s + (4.64 − 7.70i)9-s + (−0.0319 − 0.181i)11-s + (11.7 + 2.19i)12-s + (2.77 + 15.7i)16-s + (−4.86 + 8.42i)17-s + (14 − 11.3i)18-s + (−5.40 − 9.36i)19-s + (0.0638 − 0.362i)22-s + (20.6 + 12.1i)24-s + (−23.4 − 8.55i)25-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (0.870 − 0.491i)3-s + (0.766 + 0.642i)4-s + (0.986 − 0.164i)6-s + (0.500 + 0.866i)8-s + (0.515 − 0.856i)9-s + (−0.00290 − 0.0164i)11-s + (0.983 + 0.182i)12-s + (0.173 + 0.984i)16-s + (−0.286 + 0.495i)17-s + (0.777 − 0.628i)18-s + (−0.284 − 0.492i)19-s + (0.00290 − 0.0164i)22-s + (0.861 + 0.507i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.979 - 0.203i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.41199 + 0.350999i\)
\(L(\frac12)\) \(\approx\) \(3.41199 + 0.350999i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 - 0.684i)T \)
3 \( 1 + (-2.61 + 1.47i)T \)
good5 \( 1 + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (0.0319 + 0.181i)T + (-113. + 41.3i)T^{2} \)
13 \( 1 + (-129. + 108. i)T^{2} \)
17 \( 1 + (4.86 - 8.42i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (5.40 + 9.36i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-644. - 540. i)T^{2} \)
31 \( 1 + (-166. - 946. i)T^{2} \)
37 \( 1 + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (55.3 - 20.1i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-2.75 - 15.6i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-383. + 2.17e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (16.9 - 96.3i)T + (-3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-646. + 3.66e3i)T^{2} \)
67 \( 1 + (-27.1 + 9.87i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (72.5 + 125. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-115. - 42.0i)T + (5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (7.59 + 13.1i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (31.4 + 178. i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−12.38627194726227631780358397870, −11.52824659422238619076311620399, −10.22550051658225054519510022666, −8.840166540943421056147532116931, −7.962762488030194300456355734453, −6.97908876139995289451857935217, −6.02912246842940639516828693774, −4.49834202977684728535203124600, −3.33882921276189601130612263410, −2.02673570781524946794090693842, 1.99178623738137250140552640044, 3.30879107304334293412146578409, 4.34466442013350930220960743889, 5.48495709663975199848356251873, 6.89864890095990358393391583382, 8.032053510067977022424643658333, 9.334033896939316733269163961300, 10.19292313161031457866062473057, 11.12125720112288699619394058063, 12.17937791102139738638399332629

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