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Properties

Label 2-6e3-27.7-c1-0-2
Degree $2$
Conductor $216$
Sign $0.188 - 0.982i$
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  + (0.887 + 1.48i)3-s + (0.444 + 2.51i)5-s + (−0.612 − 0.513i)7-s + (−1.42 + 2.63i)9-s + (0.313 − 1.77i)11-s + (−2.44 − 0.888i)13-s + (−3.35 + 2.89i)15-s + (3.13 + 5.43i)17-s + (2.88 − 5.00i)19-s + (0.220 − 1.36i)21-s + (1.80 − 1.51i)23-s + (−1.45 + 0.528i)25-s + (−5.19 + 0.222i)27-s + (7.05 − 2.56i)29-s + (−4.55 + 3.81i)31-s + ⋯
L(s)  = 1  + (0.512 + 0.858i)3-s + (0.198 + 1.12i)5-s + (−0.231 − 0.194i)7-s + (−0.475 + 0.879i)9-s + (0.0946 − 0.536i)11-s + (−0.677 − 0.246i)13-s + (−0.865 + 0.747i)15-s + (0.760 + 1.31i)17-s + (0.662 − 1.14i)19-s + (0.0481 − 0.298i)21-s + (0.377 − 0.316i)23-s + (−0.290 + 0.105i)25-s + (−0.999 + 0.0428i)27-s + (1.31 − 0.476i)29-s + (−0.817 + 0.685i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.188 - 0.982i$
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: \((2,\ 216,\ (\ :1/2),\ 0.188 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.06981 + 0.883813i\)
\(L(\frac12)\) \(\approx\) \(1.06981 + 0.883813i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.887 - 1.48i)T \)
good5 \( 1 + (-0.444 - 2.51i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (0.612 + 0.513i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-0.313 + 1.77i)T + (-10.3 - 3.76i)T^{2} \)
13 \( 1 + (2.44 + 0.888i)T + (9.95 + 8.35i)T^{2} \)
17 \( 1 + (-3.13 - 5.43i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.88 + 5.00i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.80 + 1.51i)T + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (-7.05 + 2.56i)T + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (4.55 - 3.81i)T + (5.38 - 30.5i)T^{2} \)
37 \( 1 + (-0.0710 - 0.123i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-7.49 - 2.72i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.02 + 11.4i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.93 + 2.46i)T + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 12.3T + 53T^{2} \)
59 \( 1 + (-0.688 - 3.90i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (10.0 + 8.39i)T + (10.5 + 60.0i)T^{2} \)
67 \( 1 + (-13.7 - 5.01i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.27 + 5.67i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.483 - 0.837i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.693 - 0.252i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (6.18 - 2.25i)T + (63.5 - 53.3i)T^{2} \)
89 \( 1 + (2.90 - 5.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (0.957 - 5.43i)T + (-91.1 - 33.1i)T^{2} \)
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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

−12.57504483689732361031716747067, −11.19253327470237067421180127766, −10.51490275595068355700731457936, −9.801693942360431262807334026045, −8.686444592338594402993368080838, −7.56082134884253405957298678392, −6.40630350232032733028628958216, −5.09238858840428866563753483734, −3.59748217274914855258526080034, −2.68915013069011854075485551360, 1.31482786748747741583014245031, 2.95129293604983968274156224611, 4.72228780406298944038184190163, 5.88350235014654639357017293364, 7.25341182789921681052650926873, 8.017414852569409330142461121665, 9.298021984218293901999320074248, 9.643836977480081701438378488237, 11.51694051118419793315550475132, 12.49744190982008825775902638632

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