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Properties

Label 2-6e3-216.139-c2-0-20
Degree $2$
Conductor $216$
Sign $0.260 - 0.965i$
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  + (1.87 + 0.684i)2-s + (−2.95 − 0.493i)3-s + (3.06 + 2.57i)4-s + (−5.22 − 2.95i)6-s + (4.00 + 6.92i)8-s + (8.51 + 2.92i)9-s + (3.75 + 21.3i)11-s + (−7.79 − 9.12i)12-s + (2.77 + 15.7i)16-s + (6.74 − 11.6i)17-s + (14 + 11.3i)18-s + (11.3 + 19.5i)19-s + (−7.51 + 42.6i)22-s + (−8.41 − 22.4i)24-s + (−23.4 − 8.55i)25-s + ⋯
L(s)  = 1  + (0.939 + 0.342i)2-s + (−0.986 − 0.164i)3-s + (0.766 + 0.642i)4-s + (−0.870 − 0.491i)6-s + (0.500 + 0.866i)8-s + (0.945 + 0.324i)9-s + (0.341 + 1.93i)11-s + (−0.649 − 0.760i)12-s + (0.173 + 0.984i)16-s + (0.396 − 0.687i)17-s + (0.777 + 0.628i)18-s + (0.595 + 1.03i)19-s + (−0.341 + 1.93i)22-s + (−0.350 − 0.936i)24-s + (−0.939 − 0.342i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.260 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.260 - 0.965i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ 0.260 - 0.965i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.59450 + 1.22173i\)
\(L(\frac12)\) \(\approx\) \(1.59450 + 1.22173i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.87 - 0.684i)T \)
3 \( 1 + (2.95 + 0.493i)T \)
good5 \( 1 + (23.4 + 8.55i)T^{2} \)
7 \( 1 + (-8.50 + 48.2i)T^{2} \)
11 \( 1 + (-3.75 - 21.3i)T + (-113. + 41.3i)T^{2} \)
13 \( 1 + (-129. + 108. i)T^{2} \)
17 \( 1 + (-6.74 + 11.6i)T + (-144.5 - 250. i)T^{2} \)
19 \( 1 + (-11.3 - 19.5i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-91.8 - 520. i)T^{2} \)
29 \( 1 + (-644. - 540. i)T^{2} \)
31 \( 1 + (-166. - 946. i)T^{2} \)
37 \( 1 + (684.5 + 1.18e3i)T^{2} \)
41 \( 1 + (-70.3 + 25.5i)T + (1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (7.32 + 41.5i)T + (-1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-383. + 2.17e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (-12.0 + 68.2i)T + (-3.27e3 - 1.19e3i)T^{2} \)
61 \( 1 + (-646. + 3.66e3i)T^{2} \)
67 \( 1 + (116. - 42.3i)T + (3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (2.52e3 + 4.36e3i)T^{2} \)
73 \( 1 + (60.9 + 105. i)T + (-2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (-32.8 - 11.9i)T + (5.27e3 + 4.42e3i)T^{2} \)
89 \( 1 + (-80.5 - 139. i)T + (-3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (-6.43 - 36.5i)T + (-8.84e3 + 3.21e3i)T^{2} \)
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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

−12.07338937263999786952334107377, −11.93516008460080964689790489157, −10.53404770108246590583201256712, −9.623428725169525713575537991998, −7.68685486289544847985872335933, −7.09920316775304796037931319633, −5.96038150433376893810299221870, −4.97365103142143727610703544375, −3.98692914527065820882820828697, −1.94405684849745852986105479750, 1.02003777088971115717093615898, 3.20140842967109006895557688265, 4.39051893026324788575227025587, 5.71966907216962021590665691176, 6.17455136733857796510638406767, 7.54470515852636803140320636141, 9.197525598061013969901254015233, 10.38848975617351672764852289686, 11.26805608198800647013623350923, 11.65972687256174244270309086966

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