Properties

Label 2-18-9.2-c6-0-5
Degree $2$
Conductor $18$
Sign $0.840 + 0.541i$
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 + (−14.9 − 22.4i)3-s + (15.9 + 27.7i)4-s + (202. − 116. i)5-s + (−9.58 − 152. i)6-s + (95.5 − 165. i)7-s + 181. i·8-s + (−282. + 672. i)9-s + 1.32e3·10-s + (−673. − 388. i)11-s + (384. − 773. i)12-s + (45.5 + 78.9i)13-s + (936. − 540. i)14-s + (−5.64e3 − 2.80e3i)15-s + (−512. + 886. i)16-s + 7.04e3i·17-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (−0.553 − 0.832i)3-s + (0.249 + 0.433i)4-s + (1.61 − 0.934i)5-s + (−0.0443 − 0.705i)6-s + (0.278 − 0.482i)7-s + 0.353i·8-s + (−0.387 + 0.921i)9-s + 1.32·10-s + (−0.505 − 0.291i)11-s + (0.222 − 0.447i)12-s + (0.0207 + 0.0359i)13-s + (0.341 − 0.197i)14-s + (−1.67 − 0.830i)15-s + (−0.125 + 0.216i)16-s + 1.43i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.840 + 0.541i$
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: \((2,\ 18,\ (\ :3),\ 0.840 + 0.541i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.98741 - 0.584998i\)
\(L(\frac12)\) \(\approx\) \(1.98741 - 0.584998i\)
\(L(4)\) 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 \( 1 + (14.9 + 22.4i)T \)
good5 \( 1 + (-202. + 116. i)T + (7.81e3 - 1.35e4i)T^{2} \)
7 \( 1 + (-95.5 + 165. i)T + (-5.88e4 - 1.01e5i)T^{2} \)
11 \( 1 + (673. + 388. i)T + (8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + (-45.5 - 78.9i)T + (-2.41e6 + 4.18e6i)T^{2} \)
17 \( 1 - 7.04e3iT - 2.41e7T^{2} \)
19 \( 1 - 2.73e3T + 4.70e7T^{2} \)
23 \( 1 + (1.72e4 - 9.94e3i)T + (7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + (-2.71e4 - 1.56e4i)T + (2.97e8 + 5.15e8i)T^{2} \)
31 \( 1 + (-6.17e3 - 1.06e4i)T + (-4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + 2.79e4T + 2.56e9T^{2} \)
41 \( 1 + (3.74e4 - 2.16e4i)T + (2.37e9 - 4.11e9i)T^{2} \)
43 \( 1 + (-1.92e4 + 3.33e4i)T + (-3.16e9 - 5.47e9i)T^{2} \)
47 \( 1 + (1.43e5 + 8.30e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + 5.47e4iT - 2.21e10T^{2} \)
59 \( 1 + (1.41e4 - 8.14e3i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-2.94e4 + 5.09e4i)T + (-2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (1.47e5 + 2.56e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 1.57e5iT - 1.28e11T^{2} \)
73 \( 1 - 8.02e4T + 1.51e11T^{2} \)
79 \( 1 + (-1.88e5 + 3.26e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + (-7.33e5 - 4.23e5i)T + (1.63e11 + 2.83e11i)T^{2} \)
89 \( 1 - 1.12e3iT - 4.96e11T^{2} \)
97 \( 1 + (-6.75e5 + 1.16e6i)T + (-4.16e11 - 7.21e11i)T^{2} \)
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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

−17.29139922706735846707004970736, −16.27074516525627931739407220420, −14.03467266089762857567130013832, −13.34860981476329313118283518343, −12.28653500730699933857477124867, −10.37389348947025042140834347808, −8.252838173412415976424399969281, −6.29300855829618507063050984240, −5.19218039057125298158701569428, −1.64997347428988153635377497744, 2.60353758399536868074866720700, 5.12952550875453119040663468064, 6.34625458706468386567629492009, 9.623049250838380313456543442972, 10.47026477636543706282173826006, 11.87379381084840535983161764829, 13.70063761268388598062152130672, 14.66190269839696483640443271229, 15.99699330801760358346650753135, 17.69116123045969451275203970486

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