Properties

Label 2-18-9.2-c6-0-0
Degree $2$
Conductor $18$
Sign $-0.629 - 0.777i$
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 + (−11.1 − 24.5i)3-s + (15.9 + 27.7i)4-s + (−39.5 + 22.8i)5-s + (−14.6 + 152. i)6-s + (−245. + 424. i)7-s − 181. i·8-s + (−478. + 550. i)9-s + 258.·10-s + (−873. − 504. i)11-s + (501. − 703. i)12-s + (466. + 808. i)13-s + (2.40e3 − 1.38e3i)14-s + (1.00e3 + 716. i)15-s + (−512. + 886. i)16-s − 8.09e3i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.414 − 0.909i)3-s + (0.249 + 0.433i)4-s + (−0.316 + 0.182i)5-s + (−0.0677 + 0.703i)6-s + (−0.714 + 1.23i)7-s − 0.353i·8-s + (−0.655 + 0.754i)9-s + 0.258·10-s + (−0.656 − 0.378i)11-s + (0.290 − 0.407i)12-s + (0.212 + 0.368i)13-s + (0.875 − 0.505i)14-s + (0.297 + 0.212i)15-s + (−0.125 + 0.216i)16-s − 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.629 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $-0.629 - 0.777i$
Analytic conductor: \(4.14097\)
Root analytic conductor: \(2.03493\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :3),\ -0.629 - 0.777i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.0318399 + 0.0667645i\)
\(L(\frac12)\) \(\approx\) \(0.0318399 + 0.0667645i\)
\(L(4)\) 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 \( 1 + (11.1 + 24.5i)T \)
good5 \( 1 + (39.5 - 22.8i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (245. - 424. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (873. + 504. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-466. - 808. i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 + 8.09e3iT - 2.41e7T^{2} \)
19 \( 1 + 7.72e3T + 4.70e7T^{2} \)
23 \( 1 + (1.18e4 - 6.84e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (1.96e3 + 1.13e3i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (1.70e4 + 2.95e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 - 9.20e4T + 2.56e9T^{2} \)
41 \( 1 + (3.10e4 - 1.79e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (-3.45e4 + 5.98e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (-1.32e4 - 7.62e3i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 - 2.36e5iT - 2.21e10T^{2} \)
59 \( 1 + (2.21e5 - 1.28e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (1.99e4 - 3.45e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.60e5 + 2.77e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 4.04e5iT - 1.28e11T^{2} \)
73 \( 1 - 3.93e5T + 1.51e11T^{2} \)
79 \( 1 + (4.49e5 - 7.78e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-1.54e5 - 8.94e4i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 + 8.26e5iT - 4.96e11T^{2} \)
97 \( 1 + (3.17e5 - 5.50e5i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−18.39116073141736199074552062969, −16.70336866291349482255243499531, −15.54107748701911683921088762785, −13.44314677945778322562652032342, −12.19402719574377865925444801967, −11.20287774412839682591382636092, −9.244253685062874394402702866234, −7.67850870224326549487971886562, −5.99014583345743936276167598944, −2.51361554893894988813593547435, 0.05974136556983953401149895312, 4.11972677104944947336130834320, 6.28948962922804108803581722212, 8.208698560763989655009045311140, 10.04541249643037992735655091509, 10.73917007710819511588954036731, 12.77161868001907682824024231596, 14.69785449599607124064918014746, 15.93398224942121299746825444193, 16.75939012292823276147856828268

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