Properties

Label 2-18-9.5-c6-0-0
Degree $2$
Conductor $18$
Sign $0.836 - 0.547i$
Analytic cond. $4.14097$
Root an. cond. $2.03493$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.89 + 2.82i)2-s + (−22.5 − 14.8i)3-s + (15.9 − 27.7i)4-s + (156. + 90.6i)5-s + (152. + 9.01i)6-s + (104. + 180. i)7-s + 181. i·8-s + (287. + 669. i)9-s − 1.02e3·10-s + (2.30e3 − 1.32e3i)11-s + (−772. + 387. i)12-s + (−438. + 759. i)13-s + (−1.02e3 − 590. i)14-s + (−2.19e3 − 4.37e3i)15-s + (−512. − 886. i)16-s + 4.42e3i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.834 − 0.550i)3-s + (0.249 − 0.433i)4-s + (1.25 + 0.724i)5-s + (0.705 + 0.0417i)6-s + (0.304 + 0.526i)7-s + 0.353i·8-s + (0.394 + 0.918i)9-s − 1.02·10-s + (1.72 − 0.998i)11-s + (−0.447 + 0.223i)12-s + (−0.199 + 0.345i)13-s + (−0.372 − 0.215i)14-s + (−0.649 − 1.29i)15-s + (−0.125 − 0.216i)16-s + 0.901i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.836 - 0.547i$
Analytic conductor: \(4.14097\)
Root analytic conductor: \(2.03493\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :3),\ 0.836 - 0.547i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.06095 + 0.316580i\)
\(L(\frac12)\) \(\approx\) \(1.06095 + 0.316580i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (4.89 - 2.82i)T \)
3 \( 1 + (22.5 + 14.8i)T \)
good5 \( 1 + (-156. - 90.6i)T + (7.81e3 + 1.35e4i)T^{2} \)
7 \( 1 + (-104. - 180. i)T + (-5.88e4 + 1.01e5i)T^{2} \)
11 \( 1 + (-2.30e3 + 1.32e3i)T + (8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + (438. - 759. i)T + (-2.41e6 - 4.18e6i)T^{2} \)
17 \( 1 - 4.42e3iT - 2.41e7T^{2} \)
19 \( 1 + 4.19e3T + 4.70e7T^{2} \)
23 \( 1 + (-1.03e4 - 6.00e3i)T + (7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + (-2.28e3 + 1.32e3i)T + (2.97e8 - 5.15e8i)T^{2} \)
31 \( 1 + (2.90e3 - 5.02e3i)T + (-4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 - 4.15e4T + 2.56e9T^{2} \)
41 \( 1 + (6.47e4 + 3.73e4i)T + (2.37e9 + 4.11e9i)T^{2} \)
43 \( 1 + (7.30e4 + 1.26e5i)T + (-3.16e9 + 5.47e9i)T^{2} \)
47 \( 1 + (2.23e4 - 1.28e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 - 1.97e5iT - 2.21e10T^{2} \)
59 \( 1 + (3.13e4 + 1.80e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.19e4 + 2.07e4i)T + (-2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (1.76e5 - 3.05e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 4.96e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.82e5T + 1.51e11T^{2} \)
79 \( 1 + (1.93e5 + 3.34e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + (-3.59e5 + 2.07e5i)T + (1.63e11 - 2.83e11i)T^{2} \)
89 \( 1 - 4.05e5iT - 4.96e11T^{2} \)
97 \( 1 + (1.78e5 + 3.08e5i)T + (-4.16e11 + 7.21e11i)T^{2} \)
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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

−17.34046475468800231707097307462, −16.85070456645013304992978155377, −14.82352001146554451657898141781, −13.64117834681327302449944878316, −11.75393937354533707458213987525, −10.53048758410218736014418233441, −8.916765388414434587464367624836, −6.72073395963298167458554111107, −5.84535736133848716691370604527, −1.65792738516953793636586036853, 1.24705781352702887841057263612, 4.66882085037931787228778064770, 6.62502861487833775361075059611, 9.205319456922719782726839582936, 10.01713059621659912916558875343, 11.56735318939454563878435593924, 12.89116441520318149953780794593, 14.69779421784529223433934039204, 16.66348232388976480140557816013, 17.17871231696404850457973555412

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