Properties

Label 2-6e3-72.43-c2-0-6
Degree $2$
Conductor $216$
Sign $-0.894 - 0.447i$
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  + (0.865 + 1.80i)2-s + (−2.50 + 3.12i)4-s + (−1.70 + 0.983i)5-s + (8.69 + 5.02i)7-s + (−7.79 − 1.80i)8-s + (−3.24 − 2.22i)10-s + (−6.08 + 10.5i)11-s + (−4.28 + 2.47i)13-s + (−1.52 + 20.0i)14-s + (−3.49 − 15.6i)16-s − 4.71·17-s − 20.5·19-s + (1.18 − 7.78i)20-s + (−24.2 − 1.84i)22-s + (−3.33 + 1.92i)23-s + ⋯
L(s)  = 1  + (0.432 + 0.901i)2-s + (−0.625 + 0.780i)4-s + (−0.340 + 0.196i)5-s + (1.24 + 0.717i)7-s + (−0.974 − 0.225i)8-s + (−0.324 − 0.222i)10-s + (−0.553 + 0.958i)11-s + (−0.329 + 0.190i)13-s + (−0.108 + 1.43i)14-s + (−0.218 − 0.975i)16-s − 0.277·17-s − 1.08·19-s + (0.0594 − 0.389i)20-s + (−1.10 − 0.0837i)22-s + (−0.144 + 0.0836i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.894 - 0.447i$
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: \((2,\ 216,\ (\ :1),\ -0.894 - 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.357719 + 1.51570i\)
\(L(\frac12)\) \(\approx\) \(0.357719 + 1.51570i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.865 - 1.80i)T \)
3 \( 1 \)
good5 \( 1 + (1.70 - 0.983i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-8.69 - 5.02i)T + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (6.08 - 10.5i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (4.28 - 2.47i)T + (84.5 - 146. i)T^{2} \)
17 \( 1 + 4.71T + 289T^{2} \)
19 \( 1 + 20.5T + 361T^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-40.7 - 23.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (-49.9 + 28.8i)T + (480.5 - 832. i)T^{2} \)
37 \( 1 - 7.93iT - 1.36e3T^{2} \)
41 \( 1 + (-11.3 - 19.6i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-30.7 + 53.3i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-44.7 - 25.8i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 51.0iT - 2.80e3T^{2} \)
59 \( 1 + (16.7 + 28.9i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (39.7 + 22.9i)T + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-26.9 - 46.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 132. iT - 5.04e3T^{2} \)
73 \( 1 - 24.6T + 5.32e3T^{2} \)
79 \( 1 + (-84.3 - 48.6i)T + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (0.187 - 0.324i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 - 134.T + 7.92e3T^{2} \)
97 \( 1 + (-10.8 + 18.8i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.43446125201463046104708699808, −11.87680003002291698068346070106, −10.65254772884949722174232988785, −9.252974993066595441284206634354, −8.229316396124010095888161737253, −7.55228294878620809327574948855, −6.35506057022522856039550258775, −5.06259134536712982866257973982, −4.33694777192713056511081980775, −2.43958787759629046508088202910, 0.77564349182031075066022669226, 2.52200556029235519051553807903, 4.14689409618684415325616637895, 4.89014605868806638254288065322, 6.26612916409307233533348432499, 7.996390998684446378391016626974, 8.600144615948319944387286059504, 10.19328598678865324045213601717, 10.78521041612329449239117085339, 11.64676004889222562276446665651

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