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Properties

Label 2-18-9.2-c2-0-1
Degree $2$
Conductor $18$
Sign $0.986 - 0.164i$
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  + (1.22 + 0.707i)2-s + (−2.44 − 1.73i)3-s + (0.999 + 1.73i)4-s + (−4.5 + 2.59i)5-s + (−1.77 − 3.85i)6-s + (4.17 − 7.22i)7-s + 2.82i·8-s + (2.99 + 8.48i)9-s − 7.34·10-s + (0.825 + 0.476i)11-s + (0.550 − 5.97i)12-s + (4.84 + 8.39i)13-s + (10.2 − 5.90i)14-s + (15.5 + 1.43i)15-s + (−2.00 + 3.46i)16-s − 18.8i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.816 − 0.577i)3-s + (0.249 + 0.433i)4-s + (−0.900 + 0.519i)5-s + (−0.295 − 0.642i)6-s + (0.596 − 1.03i)7-s + 0.353i·8-s + (0.333 + 0.942i)9-s − 0.734·10-s + (0.0750 + 0.0433i)11-s + (0.0458 − 0.497i)12-s + (0.372 + 0.645i)13-s + (0.730 − 0.421i)14-s + (1.03 + 0.0953i)15-s + (−0.125 + 0.216i)16-s − 1.11i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.986 - 0.164i$
Analytic conductor: \(0.490464\)
Root analytic conductor: \(0.700331\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :1),\ 0.986 - 0.164i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.863698 + 0.0715493i\)
\(L(\frac12)\) \(\approx\) \(0.863698 + 0.0715493i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 - 0.707i)T \)
3 \( 1 + (2.44 + 1.73i)T \)
good5 \( 1 + (4.5 - 2.59i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (-4.17 + 7.22i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-0.825 - 0.476i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (-4.84 - 8.39i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 18.8iT - 289T^{2} \)
19 \( 1 + 24.6T + 361T^{2} \)
23 \( 1 + (-0.825 + 0.476i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-11.8 - 6.84i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (1.52 + 2.63i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 46.6T + 1.36e3T^{2} \)
41 \( 1 + (9.45 - 5.45i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (22.5 - 39.0i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-39.2 - 22.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 94.3iT - 2.80e3T^{2} \)
59 \( 1 + (16.2 - 9.39i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (6.54 - 11.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (37.5 + 64.9i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 18.0iT - 5.04e3T^{2} \)
73 \( 1 + 7.90T + 5.32e3T^{2} \)
79 \( 1 + (-21.8 + 37.8i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (112. + 65.1i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 145. iT - 7.92e3T^{2} \)
97 \( 1 + (-54.9 + 95.1i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−18.44141761618071372451267806966, −17.12716164867374015421238774391, −16.08765626771291622198770957025, −14.54779867993553878089108709069, −13.30198515919509624191988871080, −11.73242323252372239017402153021, −10.89429166603966416421095008930, −7.77789387497478209741735980202, −6.68753921561256918878904219742, −4.44660395771464814953646858087, 4.28355042116739765382981373077, 5.85459802292309601854744857078, 8.547550303254200435591441805699, 10.62887905274075050268212824797, 11.81568293362148607879355783074, 12.68396183205092856634498135608, 14.98884930095895838668397672561, 15.61156906069779118460823514496, 17.07151883426733925435835553088, 18.56032922514210106998866923711

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