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Properties

Label 2-6e3-72.43-c2-0-17
Degree $2$
Conductor $216$
Sign $0.899 + 0.436i$
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  + (1.99 − 0.194i)2-s + (3.92 − 0.773i)4-s + (3.84 − 2.22i)5-s + (−0.704 − 0.406i)7-s + (7.66 − 2.30i)8-s + (7.22 − 5.16i)10-s + (3.72 − 6.44i)11-s + (−18.0 + 10.4i)13-s + (−1.48 − 0.672i)14-s + (14.8 − 6.07i)16-s − 1.74·17-s + 31.7·19-s + (13.3 − 11.6i)20-s + (6.15 − 13.5i)22-s + (−6.44 + 3.72i)23-s + ⋯
L(s)  = 1  + (0.995 − 0.0971i)2-s + (0.981 − 0.193i)4-s + (0.769 − 0.444i)5-s + (−0.100 − 0.0580i)7-s + (0.957 − 0.287i)8-s + (0.722 − 0.516i)10-s + (0.338 − 0.585i)11-s + (−1.39 + 0.803i)13-s + (−0.105 − 0.0480i)14-s + (0.925 − 0.379i)16-s − 0.102·17-s + 1.67·19-s + (0.668 − 0.584i)20-s + (0.279 − 0.615i)22-s + (−0.280 + 0.161i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.899 + 0.436i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.899 + 0.436i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.899 + 0.436i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.01521 - 0.692777i\)
\(L(\frac12)\) \(\approx\) \(3.01521 - 0.692777i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.99 + 0.194i)T \)
3 \( 1 \)
good5 \( 1 + (-3.84 + 2.22i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (0.704 + 0.406i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.72 + 6.44i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (18.0 - 10.4i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 1.74T + 289T^{2} \)
19 \( 1 - 31.7T + 361T^{2} \)
23 \( 1 + (6.44 - 3.72i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (26.9 + 15.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-4.91 + 2.83i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 62.0iT - 1.36e3T^{2} \)
41 \( 1 + (2.74 + 4.74i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (22.3 - 38.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (71.1 + 41.0i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 85.0iT - 2.80e3T^{2} \)
59 \( 1 + (-21.8 - 37.8i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (61.1 + 35.2i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-9.91 - 17.1i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 69.3iT - 5.04e3T^{2} \)
73 \( 1 + 21.7T + 5.32e3T^{2} \)
79 \( 1 + (-37.1 - 21.4i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-3.53 + 6.11i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 37.1T + 7.92e3T^{2} \)
97 \( 1 + (-9.38 + 16.2i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.95360791376154269833987751193, −11.50930662721759394569335973015, −9.999142861588262660645577797881, −9.400831163112972672162324519359, −7.75157607958483178515922942731, −6.66005077409222952342669151152, −5.57157118509161740656009187588, −4.68278941458747869197859119287, −3.18081618417368861164363130600, −1.67784998112304852102317728685, 2.09336122397637110551531889662, 3.30081416189531728277039119220, 4.89385876288880739865863794028, 5.75974064753980749389443605076, 6.95109767571530461306312750760, 7.72336928114806539401570451408, 9.554061771112100829999166999239, 10.22969887929476886235563846487, 11.41579360930912884025680696093, 12.35531564303415678345981821009

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