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Properties

Label 2-6e3-216.59-c1-0-8
Degree $2$
Conductor $216$
Sign $-0.486 - 0.873i$
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  + (0.909 + 1.08i)2-s + (0.456 + 1.67i)3-s + (−0.347 + 1.96i)4-s + (−1.39 + 2.01i)6-s + (−2.44 + 1.41i)8-s + (−2.58 + 1.52i)9-s + (2.18 − 6.01i)11-s + (−3.44 + 0.317i)12-s + (−3.75 − 1.36i)16-s + (7.09 + 4.09i)17-s + (−4.00 − 1.41i)18-s + (−0.511 − 0.885i)19-s + (8.50 − 3.09i)22-s + (−3.48 − 3.44i)24-s + (−3.83 + 3.21i)25-s + ⋯
L(s)  = 1  + (0.642 + 0.766i)2-s + (0.263 + 0.964i)3-s + (−0.173 + 0.984i)4-s + (−0.569 + 0.821i)6-s + (−0.866 + 0.500i)8-s + (−0.861 + 0.507i)9-s + (0.659 − 1.81i)11-s + (−0.995 + 0.0917i)12-s + (−0.939 − 0.342i)16-s + (1.72 + 0.993i)17-s + (−0.942 − 0.333i)18-s + (−0.117 − 0.203i)19-s + (1.81 − 0.659i)22-s + (−0.710 − 0.703i)24-s + (−0.766 + 0.642i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $-0.486 - 0.873i$
Analytic conductor: \(1.72476\)
Root analytic conductor: \(1.31330\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{216} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 216,\ (\ :1/2),\ -0.486 - 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.870576 + 1.48168i\)
\(L(\frac12)\) \(\approx\) \(0.870576 + 1.48168i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.909 - 1.08i)T \)
3 \( 1 + (-0.456 - 1.67i)T \)
good5 \( 1 + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (6.57 - 2.39i)T^{2} \)
11 \( 1 + (-2.18 + 6.01i)T + (-8.42 - 7.07i)T^{2} \)
13 \( 1 + (-2.25 - 12.8i)T^{2} \)
17 \( 1 + (-7.09 - 4.09i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.511 + 0.885i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (29.1 + 10.6i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-8.15 + 9.71i)T + (-7.11 - 40.3i)T^{2} \)
43 \( 1 + (-2.07 - 0.754i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (3.84 + 10.5i)T + (-45.1 + 37.9i)T^{2} \)
61 \( 1 + (57.3 - 20.8i)T^{2} \)
67 \( 1 + (4.53 + 3.80i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.68 - 8.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-13.7 + 77.7i)T^{2} \)
83 \( 1 + (10.9 + 13.0i)T + (-14.4 + 81.7i)T^{2} \)
89 \( 1 + (11.0 - 6.38i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (14.0 + 5.12i)T + (74.3 + 62.3i)T^{2} \)
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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

−12.81664473785768302921280385514, −11.66394992909315630016896658656, −10.83198692044434170845407402044, −9.486342300852992227792493851666, −8.564019066307208717957904922872, −7.77622878214274477604102826090, −6.08861985760862356122981438412, −5.47671829736626169996222406899, −3.94602436640490176007187748870, −3.23277142599543374205074813262, 1.47049972213918293703879680815, 2.81530497815431828271566832411, 4.30848959087234503391354529459, 5.67053389201541085779608285464, 6.83955656162311161219205237048, 7.81977568006949544286182679416, 9.391901780765024804289587528027, 9.976520441584837896150176198718, 11.48590301834963993813959406614, 12.20965167024924706109042214358

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