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Properties

Label 2-6e3-72.11-c1-0-1
Degree $2$
Conductor $216$
Sign $-0.481 - 0.876i$
Analytic cond. $1.72476$
Root an. cond. $1.31330$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.608 + 1.27i)2-s + (−1.25 + 1.55i)4-s + (1.74 + 3.01i)5-s + (−1.80 − 1.04i)7-s + (−2.75 − 0.660i)8-s + (−2.79 + 4.06i)10-s + (0.116 + 0.0675i)11-s + (2.63 − 1.52i)13-s + (0.231 − 2.94i)14-s + (−0.830 − 3.91i)16-s + 4.19i·17-s + 0.919·19-s + (−6.88 − 1.09i)20-s + (−0.0150 + 0.190i)22-s + (0.689 + 1.19i)23-s + ⋯
L(s)  = 1  + (0.430 + 0.902i)2-s + (−0.629 + 0.777i)4-s + (0.779 + 1.35i)5-s + (−0.683 − 0.394i)7-s + (−0.972 − 0.233i)8-s + (−0.883 + 1.28i)10-s + (0.0352 + 0.0203i)11-s + (0.731 − 0.422i)13-s + (0.0619 − 0.786i)14-s + (−0.207 − 0.978i)16-s + 1.01i·17-s + 0.210·19-s + (−1.53 − 0.244i)20-s + (−0.00319 + 0.0406i)22-s + (0.143 + 0.249i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.739787 + 1.25076i\)
\(L(\frac12)\) \(\approx\) \(0.739787 + 1.25076i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 - 1.27i)T \)
3 \( 1 \)
good5 \( 1 + (-1.74 - 3.01i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1.80 + 1.04i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.116 - 0.0675i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.63 + 1.52i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 - 0.919T + 19T^{2} \)
23 \( 1 + (-0.689 - 1.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.24 + 7.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.39 + 2.53i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 1.61iT - 37T^{2} \)
41 \( 1 + (1.79 - 1.03i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.41 + 9.37i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.205 - 0.356i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 0.968T + 53T^{2} \)
59 \( 1 + (3.88 - 2.24i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.44 - 4.29i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.15 - 5.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 + (10.8 + 6.27i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.23 + 3.02i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 8.35iT - 89T^{2} \)
97 \( 1 + (0.477 - 0.826i)T + (-48.5 - 84.0i)T^{2} \)
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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

−13.13857416581420989580583331038, −11.77669314218874956650428543222, −10.48023082786025344927128268324, −9.824096430661501225930149247781, −8.481405899313338932023638415352, −7.30109815984933058351588412061, −6.38140451601094265177119429330, −5.81511330117344553970939017791, −3.98279407347872364338522389027, −2.86650321551862434045265358158, 1.26528598653488102301284559293, 2.91468549325536639463487465788, 4.52887054318541582885384402689, 5.43480057496570884869188736802, 6.47557465785468811448257459455, 8.576011883761197141663861084869, 9.227813609193659790363792633776, 9.936349589829490690440991753204, 11.18343756079347912098435297541, 12.26354356669023474993110142885

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