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Properties

Label 2-6e3-8.3-c2-0-4
Degree $2$
Conductor $216$
Sign $-0.994 + 0.100i$
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  + (−0.941 + 1.76i)2-s + (−2.22 − 3.32i)4-s + 8.60i·5-s + 4.81i·7-s + (7.95 − 0.805i)8-s + (−15.1 − 8.10i)10-s + 9.93·11-s − 5.78i·13-s + (−8.50 − 4.53i)14-s + (−6.06 + 14.8i)16-s − 30.8·17-s − 18.5·19-s + (28.5 − 19.1i)20-s + (−9.34 + 17.5i)22-s + 15.6i·23-s + ⋯
L(s)  = 1  + (−0.470 + 0.882i)2-s + (−0.557 − 0.830i)4-s + 1.72i·5-s + 0.688i·7-s + (0.994 − 0.100i)8-s + (−1.51 − 0.810i)10-s + 0.902·11-s − 0.445i·13-s + (−0.607 − 0.324i)14-s + (−0.379 + 0.925i)16-s − 1.81·17-s − 0.978·19-s + (1.42 − 0.958i)20-s + (−0.424 + 0.796i)22-s + 0.682i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.994 + 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.994 + 0.100i$
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: \((2,\ 216,\ (\ :1),\ -0.994 + 0.100i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0429544 - 0.850560i\)
\(L(\frac12)\) \(\approx\) \(0.0429544 - 0.850560i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.941 - 1.76i)T \)
3 \( 1 \)
good5 \( 1 - 8.60iT - 25T^{2} \)
7 \( 1 - 4.81iT - 49T^{2} \)
11 \( 1 - 9.93T + 121T^{2} \)
13 \( 1 + 5.78iT - 169T^{2} \)
17 \( 1 + 30.8T + 289T^{2} \)
19 \( 1 + 18.5T + 361T^{2} \)
23 \( 1 - 15.6iT - 529T^{2} \)
29 \( 1 - 25.9iT - 841T^{2} \)
31 \( 1 - 12.7iT - 961T^{2} \)
37 \( 1 + 63.2iT - 1.36e3T^{2} \)
41 \( 1 - 14.4T + 1.68e3T^{2} \)
43 \( 1 - 62.4T + 1.84e3T^{2} \)
47 \( 1 - 26.7iT - 2.20e3T^{2} \)
53 \( 1 + 14.2iT - 2.80e3T^{2} \)
59 \( 1 + 76.7T + 3.48e3T^{2} \)
61 \( 1 - 44.5iT - 3.72e3T^{2} \)
67 \( 1 - 88.6T + 4.48e3T^{2} \)
71 \( 1 - 114. iT - 5.04e3T^{2} \)
73 \( 1 - 28.3T + 5.32e3T^{2} \)
79 \( 1 - 115. iT - 6.24e3T^{2} \)
83 \( 1 - 46.2T + 6.88e3T^{2} \)
89 \( 1 + 33.8T + 7.92e3T^{2} \)
97 \( 1 + 93.6T + 9.40e3T^{2} \)
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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

−12.69884705435509884159557713353, −11.12382541520963859001546940451, −10.76010092124220099379353257887, −9.465627798857719770093368435823, −8.660039694266738930315766127726, −7.30407629796070184729503269212, −6.61812858352266631694359405634, −5.77833875298333977690411783080, −4.04928177700476229796458933988, −2.31569622755208168264810679896, 0.55141016939541994111013206646, 1.93900862281256485175329293854, 4.20003250650028021720178662569, 4.54384675661120075537927241308, 6.51978074126349978059829748896, 8.023280683293302088752476000297, 8.884838199573736751460140591844, 9.412835106543348580981906941185, 10.68317509820526304592618218169, 11.63734862565400932828476925115

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