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Properties

Label 2-6e3-27.13-c1-0-5
Degree $2$
Conductor $216$
Sign $0.991 + 0.129i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
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  + (1.59 − 0.678i)3-s + (1.47 + 1.23i)5-s + (−0.495 − 0.180i)7-s + (2.07 − 2.16i)9-s + (−1.04 + 0.877i)11-s + (−0.231 + 1.31i)13-s + (3.18 + 0.970i)15-s + (−1.45 + 2.51i)17-s + (−4.12 − 7.14i)19-s + (−0.912 + 0.0489i)21-s + (4.64 − 1.68i)23-s + (−0.226 − 1.28i)25-s + (1.84 − 4.85i)27-s + (1.56 + 8.84i)29-s + (−7.35 + 2.67i)31-s + ⋯
L(s)  = 1  + (0.920 − 0.391i)3-s + (0.658 + 0.552i)5-s + (−0.187 − 0.0682i)7-s + (0.692 − 0.720i)9-s + (−0.315 + 0.264i)11-s + (−0.0642 + 0.364i)13-s + (0.822 + 0.250i)15-s + (−0.352 + 0.609i)17-s + (−0.946 − 1.63i)19-s + (−0.199 + 0.0106i)21-s + (0.968 − 0.352i)23-s + (−0.0452 − 0.256i)25-s + (0.355 − 0.934i)27-s + (0.289 + 1.64i)29-s + (−1.32 + 0.481i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.991 + 0.129i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ 0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66768 - 0.108765i\)
\(L(\frac12)\) \(\approx\) \(1.66768 - 0.108765i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 + 0.678i)T \)
good5 \( 1 + (-1.47 - 1.23i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (0.495 + 0.180i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.04 - 0.877i)T + (1.91 - 10.8i)T^{2} \)
13 \( 1 + (0.231 - 1.31i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.45 - 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.12 + 7.14i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.64 + 1.68i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.56 - 8.84i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (7.35 - 2.67i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (-0.567 + 0.982i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.675 - 3.82i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (1.69 - 1.42i)T + (7.46 - 42.3i)T^{2} \)
47 \( 1 + (6.47 + 2.35i)T + (36.0 + 30.2i)T^{2} \)
53 \( 1 + 8.26T + 53T^{2} \)
59 \( 1 + (-1.66 - 1.39i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-7.98 - 2.90i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.319 - 1.80i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-3.51 + 6.08i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.42 + 4.19i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.99 - 11.3i)T + (-74.2 + 27.0i)T^{2} \)
83 \( 1 + (-2.49 - 14.1i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (7.50 + 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.2 + 11.9i)T + (16.8 - 95.5i)T^{2} \)
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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

−12.83688273988482876435342527795, −11.18232307179382561186203719790, −10.29202307536480720750292133261, −9.226296295950686943348120412616, −8.487656245273887373320588956474, −7.03829917120920588301546329278, −6.54478126673060795166743810268, −4.77320392060555898554880383653, −3.18168243083638922621623144342, −2.02713753019896064978472295823, 2.01946501410179897552003953713, 3.48987839982499849822802240401, 4.86065889017900067038949664733, 6.01518724854222145459921785964, 7.57179157643303673696449177167, 8.493045394700433921808003274077, 9.431839291617585626780794909424, 10.10494522706087022778378328037, 11.23429673298961837259725500023, 12.75193344158231759367497521942

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