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Properties

Label 2-6e3-8.3-c2-0-27
Degree $2$
Conductor $216$
Sign $0.807 + 0.590i$
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  + (1.95 + 0.417i)2-s + (3.65 + 1.63i)4-s − 5.14i·5-s − 12.6i·7-s + (6.45 + 4.72i)8-s + (2.14 − 10.0i)10-s − 1.46·11-s + 5.68i·13-s + (5.28 − 24.7i)14-s + (10.6 + 11.9i)16-s − 14.2·17-s + 26.6·19-s + (8.40 − 18.7i)20-s + (−2.86 − 0.611i)22-s + 36.7i·23-s + ⋯
L(s)  = 1  + (0.977 + 0.208i)2-s + (0.912 + 0.408i)4-s − 1.02i·5-s − 1.80i·7-s + (0.807 + 0.590i)8-s + (0.214 − 1.00i)10-s − 0.132·11-s + 0.437i·13-s + (0.377 − 1.76i)14-s + (0.666 + 0.745i)16-s − 0.839·17-s + 1.40·19-s + (0.420 − 0.938i)20-s + (−0.130 − 0.0277i)22-s + 1.59i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.807 + 0.590i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.807 + 0.590i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.63403 - 0.860118i\)
\(L(\frac12)\) \(\approx\) \(2.63403 - 0.860118i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.95 - 0.417i)T \)
3 \( 1 \)
good5 \( 1 + 5.14iT - 25T^{2} \)
7 \( 1 + 12.6iT - 49T^{2} \)
11 \( 1 + 1.46T + 121T^{2} \)
13 \( 1 - 5.68iT - 169T^{2} \)
17 \( 1 + 14.2T + 289T^{2} \)
19 \( 1 - 26.6T + 361T^{2} \)
23 \( 1 - 36.7iT - 529T^{2} \)
29 \( 1 - 19.4iT - 841T^{2} \)
31 \( 1 - 16.1iT - 961T^{2} \)
37 \( 1 + 37.2iT - 1.36e3T^{2} \)
41 \( 1 + 58.8T + 1.68e3T^{2} \)
43 \( 1 - 61.2T + 1.84e3T^{2} \)
47 \( 1 + 61.6iT - 2.20e3T^{2} \)
53 \( 1 - 42.0iT - 2.80e3T^{2} \)
59 \( 1 - 2.74T + 3.48e3T^{2} \)
61 \( 1 - 71.6iT - 3.72e3T^{2} \)
67 \( 1 + 10.3T + 4.48e3T^{2} \)
71 \( 1 - 70.6iT - 5.04e3T^{2} \)
73 \( 1 + 104.T + 5.32e3T^{2} \)
79 \( 1 + 82.0iT - 6.24e3T^{2} \)
83 \( 1 - 68.7T + 6.88e3T^{2} \)
89 \( 1 - 65.6T + 7.92e3T^{2} \)
97 \( 1 - 5.31T + 9.40e3T^{2} \)
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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

−12.13903841520112920416210784041, −11.23398761638884139102461603673, −10.26652184927572506725139671965, −8.960491731343954249685223223859, −7.56403907776869953718016059151, −7.00522055606583199568317477625, −5.44232149581277881086554429959, −4.49343853068908374932327920738, −3.53777223326537198520933000398, −1.32760381152134188864969208891, 2.36246420892651853334154873370, 3.09060556709166120658879095052, 4.84632379600059397329567922749, 5.92464536278809390308531478926, 6.70166903589445741328371516823, 8.066260816864832660776391695721, 9.422199283041154777867665741887, 10.54446806424361168863155975263, 11.45717674066918220609761060423, 12.14544662819422912512553615435

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