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Properties

Label 2-6e3-72.43-c2-0-9
Degree $2$
Conductor $216$
Sign $0.771 - 0.635i$
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 + 1.73i)2-s + (−1.99 − 3.46i)4-s + 7.99·8-s + (3.84 − 6.66i)11-s + (−8 + 13.8i)16-s + 30.3·17-s + 31.6·19-s + (7.69 + 13.3i)22-s + (−12.5 + 21.6i)25-s + (−15.9 − 27.7i)32-s + (−30.3 + 52.6i)34-s + (−31.6 + 54.9i)38-s + (40.8 + 70.8i)41-s + (40.2 − 69.7i)43-s − 30.7·44-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 0.999·8-s + (0.349 − 0.605i)11-s + (−0.5 + 0.866i)16-s + 1.78·17-s + 1.66·19-s + (0.349 + 0.605i)22-s + (−0.5 + 0.866i)25-s + (−0.499 − 0.866i)32-s + (−0.893 + 1.54i)34-s + (−0.834 + 1.44i)38-s + (0.997 + 1.72i)41-s + (0.935 − 1.62i)43-s − 0.699·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.771 - 0.635i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.771 - 0.635i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(216\)    =    \(2^{3} \cdot 3^{3}\)
Sign: $0.771 - 0.635i$
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: \((2,\ 216,\ (\ :1),\ 0.771 - 0.635i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.14559 + 0.410984i\)
\(L(\frac12)\) \(\approx\) \(1.14559 + 0.410984i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
3 \( 1 \)
good5 \( 1 + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (24.5 + 42.4i)T^{2} \)
11 \( 1 + (-3.84 + 6.66i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 + (84.5 - 146. i)T^{2} \)
17 \( 1 - 30.3T + 289T^{2} \)
19 \( 1 - 31.6T + 361T^{2} \)
23 \( 1 + (264.5 - 458. i)T^{2} \)
29 \( 1 + (420.5 + 728. i)T^{2} \)
31 \( 1 + (480.5 - 832. i)T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + (-40.8 - 70.8i)T + (-840.5 + 1.45e3i)T^{2} \)
43 \( 1 + (-40.2 + 69.7i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 2.80e3T^{2} \)
59 \( 1 + (16.2 + 28.1i)T + (-1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (35.9 + 62.2i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 41.6T + 5.32e3T^{2} \)
79 \( 1 + (3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + (-79 + 136. i)T + (-3.44e3 - 5.96e3i)T^{2} \)
89 \( 1 + 146T + 7.92e3T^{2} \)
97 \( 1 + (96.9 - 167. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−12.12754988380023928746926779543, −11.10685230598203753622458360440, −9.913478507340779428676369433794, −9.272686498617297530516835828918, −8.024841636888128569183503125444, −7.33145222187899557169835312468, −5.99245713687831300836265771623, −5.19427117847164033546789147460, −3.50128333322726145194408794742, −1.11253783113985404478538028219, 1.19224508629314477858381693095, 2.88931258865112784245262770458, 4.14578394285584096654554668804, 5.56110220437488905844570875122, 7.29925071790697913586880597815, 8.030410553084850833720197251104, 9.415291492540455671202448979941, 9.894146257341043690336511474429, 11.01397518952131716408851667839, 12.07098582826271340618851945015

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