Properties

Label 2-6e3-27.7-c1-0-0
Degree $2$
Conductor $216$
Sign $-0.217 - 0.976i$
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  + (−0.631 + 1.61i)3-s + (0.156 + 0.889i)5-s + (1.02 + 0.860i)7-s + (−2.20 − 2.03i)9-s + (−1.03 + 5.84i)11-s + (−0.904 − 0.329i)13-s + (−1.53 − 0.308i)15-s + (0.115 + 0.200i)17-s + (−0.756 + 1.31i)19-s + (−2.03 + 1.11i)21-s + (−3.42 + 2.87i)23-s + (3.93 − 1.43i)25-s + (4.67 − 2.26i)27-s + (5.21 − 1.89i)29-s + (7.24 − 6.07i)31-s + ⋯
L(s)  = 1  + (−0.364 + 0.931i)3-s + (0.0701 + 0.397i)5-s + (0.387 + 0.325i)7-s + (−0.734 − 0.678i)9-s + (−0.310 + 1.76i)11-s + (−0.250 − 0.0913i)13-s + (−0.395 − 0.0795i)15-s + (0.0280 + 0.0486i)17-s + (−0.173 + 0.300i)19-s + (−0.444 + 0.242i)21-s + (−0.714 + 0.599i)23-s + (0.786 − 0.286i)25-s + (0.899 − 0.436i)27-s + (0.967 − 0.352i)29-s + (1.30 − 1.09i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.217 - 0.976i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.217 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.640192 + 0.798424i\)
\(L(\frac12)\) \(\approx\) \(0.640192 + 0.798424i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.631 - 1.61i)T \)
good5 \( 1 + (-0.156 - 0.889i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.02 - 0.860i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (1.03 - 5.84i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (0.904 + 0.329i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-0.115 - 0.200i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.756 - 1.31i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.42 - 2.87i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-5.21 + 1.89i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-7.24 + 6.07i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-1.74 - 3.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.06 + 1.84i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-1.47 + 8.38i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-5.18 - 4.35i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 - 5.69T + 53T^{2} \)
59 \( 1 + (-0.506 - 2.87i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-10.2 - 8.61i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (11.6 + 4.23i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (7.56 + 13.1i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.83 - 4.90i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.84 + 2.49i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (5.91 - 2.15i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (-7.28 + 12.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.69 - 15.2i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.20616553433182661518785503427, −11.75092227499372318592968937484, −10.28663603282672537443608348071, −10.07850422439225760463891345757, −8.810451440007366371592916106417, −7.57751436139836440151430706397, −6.32556178328133914092980438580, −5.08781612970332839855998558863, −4.18571752321924125519233282145, −2.47287868676903913212799970760, 0.961102848968339201770734970429, 2.86390908501375572705387396689, 4.77632142279650691840476261536, 5.89092406405550537260473419319, 6.88226761467549455068209031417, 8.178096495814179200644325584600, 8.672104748733841264725087247558, 10.38175572873517963945373861176, 11.19390405680200889544991271448, 12.06103976873193492650574916026

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