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Properties

Label 2-6e3-8.3-c2-0-24
Degree $2$
Conductor $216$
Sign $-0.948 - 0.317i$
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  + (−0.808 − 1.82i)2-s + (−2.69 + 2.95i)4-s + 5.51i·5-s − 11.2i·7-s + (7.58 + 2.53i)8-s + (10.0 − 4.45i)10-s − 17.9·11-s − 5.65i·13-s + (−20.5 + 9.06i)14-s + (−1.49 − 15.9i)16-s − 9.53·17-s − 33.1·19-s + (−16.3 − 14.8i)20-s + (14.4 + 32.7i)22-s − 19.3i·23-s + ⋯
L(s)  = 1  + (−0.404 − 0.914i)2-s + (−0.673 + 0.739i)4-s + 1.10i·5-s − 1.60i·7-s + (0.948 + 0.317i)8-s + (1.00 − 0.445i)10-s − 1.62·11-s − 0.435i·13-s + (−1.46 + 0.647i)14-s + (−0.0932 − 0.995i)16-s − 0.561·17-s − 1.74·19-s + (−0.815 − 0.742i)20-s + (0.658 + 1.49i)22-s − 0.843i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 - 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.948 - 0.317i$
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: \((2,\ 216,\ (\ :1),\ -0.948 - 0.317i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0464189 + 0.285246i\)
\(L(\frac12)\) \(\approx\) \(0.0464189 + 0.285246i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.808 + 1.82i)T \)
3 \( 1 \)
good5 \( 1 - 5.51iT - 25T^{2} \)
7 \( 1 + 11.2iT - 49T^{2} \)
11 \( 1 + 17.9T + 121T^{2} \)
13 \( 1 + 5.65iT - 169T^{2} \)
17 \( 1 + 9.53T + 289T^{2} \)
19 \( 1 + 33.1T + 361T^{2} \)
23 \( 1 + 19.3iT - 529T^{2} \)
29 \( 1 - 29.8iT - 841T^{2} \)
31 \( 1 - 6.52iT - 961T^{2} \)
37 \( 1 + 33.9iT - 1.36e3T^{2} \)
41 \( 1 + 56.8T + 1.68e3T^{2} \)
43 \( 1 - 19.1T + 1.84e3T^{2} \)
47 \( 1 + 30.9iT - 2.20e3T^{2} \)
53 \( 1 - 11.2iT - 2.80e3T^{2} \)
59 \( 1 - 10.2T + 3.48e3T^{2} \)
61 \( 1 + 3.18iT - 3.72e3T^{2} \)
67 \( 1 - 13.1T + 4.48e3T^{2} \)
71 \( 1 + 103. iT - 5.04e3T^{2} \)
73 \( 1 + 21.3T + 5.32e3T^{2} \)
79 \( 1 - 134. iT - 6.24e3T^{2} \)
83 \( 1 - 56.2T + 6.88e3T^{2} \)
89 \( 1 - 114.T + 7.92e3T^{2} \)
97 \( 1 - 126.T + 9.40e3T^{2} \)
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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

−10.97140036731736640819543531828, −10.60228015347447936541201617782, −10.23746481671025072091160152355, −8.573968577292897508109209179074, −7.59775431067186978882926861534, −6.74700574625841316159083708287, −4.77405133109222462558833219451, −3.56192911524437829400791507473, −2.36843477378556177191558802291, −0.17160967954942949485259395562, 2.14981205354621892680637604589, 4.63048878194196769056496267605, 5.40538604022760127324075306982, 6.35013514148755354007607080615, 7.957190633794978507773085143545, 8.591667040926757507130100473995, 9.283397740157404989564475095495, 10.42721829005800258771981672671, 11.76008247312128989351671292006, 12.92794944474491137655203856638

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