Properties

Label 2-6e3-216.101-c2-0-27
Degree $2$
Conductor $216$
Sign $-0.309 - 0.950i$
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  + (1.80 + 0.858i)2-s + (−0.216 + 2.99i)3-s + (2.52 + 3.10i)4-s + (−0.714 − 4.05i)5-s + (−2.95 + 5.21i)6-s + (5.27 + 4.42i)7-s + (1.90 + 7.76i)8-s + (−8.90 − 1.29i)9-s + (2.18 − 7.93i)10-s + (−3.61 + 20.5i)11-s + (−9.82 + 6.89i)12-s + (3.82 − 10.5i)13-s + (5.73 + 12.5i)14-s + (12.2 − 1.26i)15-s + (−3.22 + 15.6i)16-s + (17.5 − 10.1i)17-s + ⋯
L(s)  = 1  + (0.903 + 0.429i)2-s + (−0.0722 + 0.997i)3-s + (0.631 + 0.775i)4-s + (−0.142 − 0.810i)5-s + (−0.493 + 0.869i)6-s + (0.753 + 0.632i)7-s + (0.238 + 0.971i)8-s + (−0.989 − 0.144i)9-s + (0.218 − 0.793i)10-s + (−0.328 + 1.86i)11-s + (−0.818 + 0.574i)12-s + (0.294 − 0.808i)13-s + (0.409 + 0.894i)14-s + (0.818 − 0.0840i)15-s + (−0.201 + 0.979i)16-s + (1.03 − 0.595i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.309 - 0.950i$
Analytic conductor: \(5.88557\)
Root analytic conductor: \(2.42602\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1),\ -0.309 - 0.950i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.48564 + 2.04548i\)
\(L(\frac12)\) \(\approx\) \(1.48564 + 2.04548i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.80 - 0.858i)T \)
3 \( 1 + (0.216 - 2.99i)T \)
good5 \( 1 + (0.714 + 4.05i)T + (-23.4 + 8.55i)T^{2} \)
7 \( 1 + (-5.27 - 4.42i)T + (8.50 + 48.2i)T^{2} \)
11 \( 1 + (3.61 - 20.5i)T + (-113. - 41.3i)T^{2} \)
13 \( 1 + (-3.82 + 10.5i)T + (-129. - 108. i)T^{2} \)
17 \( 1 + (-17.5 + 10.1i)T + (144.5 - 250. i)T^{2} \)
19 \( 1 + (25.0 + 14.4i)T + (180.5 + 312. i)T^{2} \)
23 \( 1 + (-10.6 - 12.6i)T + (-91.8 + 520. i)T^{2} \)
29 \( 1 + (40.4 - 14.7i)T + (644. - 540. i)T^{2} \)
31 \( 1 + (-25.1 + 21.1i)T + (166. - 946. i)T^{2} \)
37 \( 1 + (-37.0 + 21.3i)T + (684.5 - 1.18e3i)T^{2} \)
41 \( 1 + (-2.39 + 6.57i)T + (-1.28e3 - 1.08e3i)T^{2} \)
43 \( 1 + (-28.3 - 4.99i)T + (1.73e3 + 632. i)T^{2} \)
47 \( 1 + (-24.4 + 29.1i)T + (-383. - 2.17e3i)T^{2} \)
53 \( 1 - 91.9T + 2.80e3T^{2} \)
59 \( 1 + (3.40 + 19.2i)T + (-3.27e3 + 1.19e3i)T^{2} \)
61 \( 1 + (22.8 - 27.1i)T + (-646. - 3.66e3i)T^{2} \)
67 \( 1 + (-38.4 + 105. i)T + (-3.43e3 - 2.88e3i)T^{2} \)
71 \( 1 + (74.4 - 42.9i)T + (2.52e3 - 4.36e3i)T^{2} \)
73 \( 1 + (12.0 - 20.9i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (25.6 - 9.33i)T + (4.78e3 - 4.01e3i)T^{2} \)
83 \( 1 + (16.7 - 6.09i)T + (5.27e3 - 4.42e3i)T^{2} \)
89 \( 1 + (81.2 + 46.9i)T + (3.96e3 + 6.85e3i)T^{2} \)
97 \( 1 + (3.52 - 19.9i)T + (-8.84e3 - 3.21e3i)T^{2} \)
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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

−12.45152352889436304873075354763, −11.62639186039781126589463191623, −10.60381503569311046537366735715, −9.343430798680534355447472384468, −8.357165756493919037526713297546, −7.35738903183979329595012566549, −5.62151900369475328897751896215, −4.97438686532241479721265713703, −4.16573081913083397141025669040, −2.46975212978126266768498929072, 1.20256129169709882728078044535, 2.77679620508534358343177612014, 3.99368997516816483526781937837, 5.72412988345069534062372456188, 6.42586366771347166616018018024, 7.56096173894137310868887538024, 8.554623702158203221838025774408, 10.54310959422240048755926578021, 11.01611154839945038859887463657, 11.73945584500768269015345474716

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