Properties

Label 2-18-3.2-c2-0-1
Degree $2$
Conductor $18$
Sign $0.816 + 0.577i$
Analytic cond. $0.490464$
Root an. cond. $0.700331$
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  − 1.41i·2-s − 2.00·4-s + 4.24i·5-s − 4·7-s + 2.82i·8-s + 6·10-s − 16.9i·11-s + 8·13-s + 5.65i·14-s + 4.00·16-s + 12.7i·17-s − 16·19-s − 8.48i·20-s − 24·22-s + 16.9i·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.500·4-s + 0.848i·5-s − 0.571·7-s + 0.353i·8-s + 0.600·10-s − 1.54i·11-s + 0.615·13-s + 0.404i·14-s + 0.250·16-s + 0.748i·17-s − 0.842·19-s − 0.424i·20-s − 1.09·22-s + 0.737i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.816 + 0.577i$
Analytic conductor: \(0.490464\)
Root analytic conductor: \(0.700331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :1),\ 0.816 + 0.577i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.751341 - 0.238804i\)
\(L(\frac12)\) \(\approx\) \(0.751341 - 0.238804i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
3 \( 1 \)
good5 \( 1 - 4.24iT - 25T^{2} \)
7 \( 1 + 4T + 49T^{2} \)
11 \( 1 + 16.9iT - 121T^{2} \)
13 \( 1 - 8T + 169T^{2} \)
17 \( 1 - 12.7iT - 289T^{2} \)
19 \( 1 + 16T + 361T^{2} \)
23 \( 1 - 16.9iT - 529T^{2} \)
29 \( 1 + 4.24iT - 841T^{2} \)
31 \( 1 - 44T + 961T^{2} \)
37 \( 1 + 34T + 1.36e3T^{2} \)
41 \( 1 + 46.6iT - 1.68e3T^{2} \)
43 \( 1 + 40T + 1.84e3T^{2} \)
47 \( 1 - 84.8iT - 2.20e3T^{2} \)
53 \( 1 + 38.1iT - 2.80e3T^{2} \)
59 \( 1 + 33.9iT - 3.48e3T^{2} \)
61 \( 1 - 50T + 3.72e3T^{2} \)
67 \( 1 - 8T + 4.48e3T^{2} \)
71 \( 1 - 50.9iT - 5.04e3T^{2} \)
73 \( 1 + 16T + 5.32e3T^{2} \)
79 \( 1 + 76T + 6.24e3T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 + 12.7iT - 7.92e3T^{2} \)
97 \( 1 - 176T + 9.40e3T^{2} \)
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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

−18.89347273255062280476635091455, −17.34631555644508276461662797034, −15.80362893975988818583436490781, −14.22146059509573790071482964497, −13.07300224456653051959814522774, −11.36515219725111670229259044791, −10.34383778911769437279554643238, −8.550251627536126227175555967214, −6.24803028236882973923729368575, −3.36175297440665047564543646315, 4.71803328341868101997909002331, 6.74703699913736062741911446047, 8.564573240899659569950682875282, 9.955810945311239495961961241930, 12.28179795767312081834728189015, 13.34283436675829660838174997309, 14.98901922992886842054569501792, 16.12940553476581195592109514968, 17.17386069467369261541271057371, 18.41989986415510726525675866418

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