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Properties

Label 2-18-9.2-c4-0-0
Degree $2$
Conductor $18$
Sign $-0.762 - 0.647i$
Analytic cond. $1.86065$
Root an. cond. $1.36405$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.44 − 1.41i)2-s + (−7.80 + 4.47i)3-s + (3.99 + 6.92i)4-s + (−32.5 + 18.7i)5-s + (25.4 + 0.0758i)6-s + (−1.35 + 2.34i)7-s − 22.6i·8-s + (40.9 − 69.9i)9-s + 106.·10-s + (−91.4 − 52.8i)11-s + (−62.2 − 36.1i)12-s + (97.9 + 169. i)13-s + (6.63 − 3.83i)14-s + (169. − 292. i)15-s + (−32.0 + 55.4i)16-s + 448. i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.867 + 0.497i)3-s + (0.249 + 0.433i)4-s + (−1.30 + 0.750i)5-s + (0.707 + 0.00210i)6-s + (−0.0276 + 0.0478i)7-s − 0.353i·8-s + (0.505 − 0.863i)9-s + 1.06·10-s + (−0.756 − 0.436i)11-s + (−0.432 − 0.251i)12-s + (0.579 + 1.00i)13-s + (0.0338 − 0.0195i)14-s + (0.754 − 1.29i)15-s + (−0.125 + 0.216i)16-s + 1.55i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.762 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $-0.762 - 0.647i$
Analytic conductor: \(1.86065\)
Root analytic conductor: \(1.36405\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :2),\ -0.762 - 0.647i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.0997432 + 0.271522i\)
\(L(\frac12)\) \(\approx\) \(0.0997432 + 0.271522i\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.44 + 1.41i)T \)
3 \( 1 + (7.80 - 4.47i)T \)
good5 \( 1 + (32.5 - 18.7i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (1.35 - 2.34i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (91.4 + 52.8i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (-97.9 - 169. i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 - 448. iT - 8.35e4T^{2} \)
19 \( 1 + 501.T + 1.30e5T^{2} \)
23 \( 1 + (-343. + 198. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (-466. - 269. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (297. + 515. i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 + 1.84e3T + 1.87e6T^{2} \)
41 \( 1 + (145. - 83.9i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (310. - 537. i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (651. + 376. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 - 913. iT - 7.89e6T^{2} \)
59 \( 1 + (-1.14e3 + 659. i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (-2.06e3 + 3.57e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (-3.34e3 - 5.78e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 + 2.88e3iT - 2.54e7T^{2} \)
73 \( 1 - 118.T + 2.83e7T^{2} \)
79 \( 1 + (3.91e3 - 6.78e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-4.40e3 - 2.54e3i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.02e4iT - 6.27e7T^{2} \)
97 \( 1 + (-440. + 762. i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−18.65336355580116895504404706038, −17.13749705989001496586322429216, −15.99011299938552794119338043089, −15.01272396766518432605816978045, −12.54846330465163390207534788885, −11.19762182619789014727868255377, −10.58257779619198657343721503436, −8.460753288661748891431913376879, −6.63692292576685849671664090014, −3.91116000143257416537261368829, 0.35896396520025872083368777390, 5.06855817932564334022882027353, 7.19077512983447310819691217662, 8.386470627778484052063895364003, 10.59000112693685631634310831085, 11.88333079410378479230525637281, 13.08589112422085762884385490645, 15.47174465584957319815862863318, 16.17232820206227241024900613930, 17.44314304780314007451040536144

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