Properties

Label 2-18-9.2-c4-0-2
Degree $2$
Conductor $18$
Sign $0.999 + 0.0124i$
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 + (6.85 + 5.82i)3-s + (3.99 + 6.92i)4-s + (37.0 − 21.3i)5-s + (−8.55 − 23.9i)6-s + (−19.8 + 34.3i)7-s − 22.6i·8-s + (13.0 + 79.9i)9-s − 120.·10-s + (−139. − 80.7i)11-s + (−12.9 + 70.8i)12-s + (−36.6 − 63.4i)13-s + (97.2 − 56.1i)14-s + (378. + 69.1i)15-s + (−32.0 + 55.4i)16-s − 65.0i·17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.762 + 0.647i)3-s + (0.249 + 0.433i)4-s + (1.48 − 0.854i)5-s + (−0.237 − 0.665i)6-s + (−0.404 + 0.701i)7-s − 0.353i·8-s + (0.161 + 0.986i)9-s − 1.20·10-s + (−1.15 − 0.667i)11-s + (−0.0898 + 0.491i)12-s + (−0.216 − 0.375i)13-s + (0.495 − 0.286i)14-s + (1.68 + 0.307i)15-s + (−0.125 + 0.216i)16-s − 0.224i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0124i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 + 0.0124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.999 + 0.0124i$
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: \((2,\ 18,\ (\ :2),\ 0.999 + 0.0124i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.26360 - 0.00789693i\)
\(L(\frac12)\) \(\approx\) \(1.26360 - 0.00789693i\)
\(L(3)\) 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 \( 1 + (-6.85 - 5.82i)T \)
good5 \( 1 + (-37.0 + 21.3i)T + (312.5 - 541. i)T^{2} \)
7 \( 1 + (19.8 - 34.3i)T + (-1.20e3 - 2.07e3i)T^{2} \)
11 \( 1 + (139. + 80.7i)T + (7.32e3 + 1.26e4i)T^{2} \)
13 \( 1 + (36.6 + 63.4i)T + (-1.42e4 + 2.47e4i)T^{2} \)
17 \( 1 + 65.0iT - 8.35e4T^{2} \)
19 \( 1 + 163.T + 1.30e5T^{2} \)
23 \( 1 + (185. - 107. i)T + (1.39e5 - 2.42e5i)T^{2} \)
29 \( 1 + (563. + 325. i)T + (3.53e5 + 6.12e5i)T^{2} \)
31 \( 1 + (-633. - 1.09e3i)T + (-4.61e5 + 7.99e5i)T^{2} \)
37 \( 1 - 278.T + 1.87e6T^{2} \)
41 \( 1 + (-48.2 + 27.8i)T + (1.41e6 - 2.44e6i)T^{2} \)
43 \( 1 + (-932. + 1.61e3i)T + (-1.70e6 - 2.96e6i)T^{2} \)
47 \( 1 + (-1.10e3 - 636. i)T + (2.43e6 + 4.22e6i)T^{2} \)
53 \( 1 + 3.58e3iT - 7.89e6T^{2} \)
59 \( 1 + (-2.93e3 + 1.69e3i)T + (6.05e6 - 1.04e7i)T^{2} \)
61 \( 1 + (3.49e3 - 6.04e3i)T + (-6.92e6 - 1.19e7i)T^{2} \)
67 \( 1 + (958. + 1.66e3i)T + (-1.00e7 + 1.74e7i)T^{2} \)
71 \( 1 - 4.58e3iT - 2.54e7T^{2} \)
73 \( 1 - 9.91e3T + 2.83e7T^{2} \)
79 \( 1 + (974. - 1.68e3i)T + (-1.94e7 - 3.37e7i)T^{2} \)
83 \( 1 + (-1.19e3 - 688. i)T + (2.37e7 + 4.11e7i)T^{2} \)
89 \( 1 - 1.37e4iT - 6.27e7T^{2} \)
97 \( 1 + (-4.14e3 + 7.18e3i)T + (-4.42e7 - 7.66e7i)T^{2} \)
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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

−17.99056415480924181200363341213, −16.66220687114557178499390265274, −15.60927200947669009841076360068, −13.76462942885900577143963963955, −12.75350423828468765095059092499, −10.44296925859278060277533966493, −9.427511489645825418478135922192, −8.392897795354847585846403299203, −5.46950545142362500623511304573, −2.48999694692836780762156599905, 2.28589222508502348599037561037, 6.31380721354533049090847999039, 7.53858231447169443503710439509, 9.491684936447435553934235958703, 10.45023962820383349089138069180, 13.02427860822468016834152867658, 14.00572588854808056094326507741, 15.13384528124555762245357909419, 17.04512993358969649783780287610, 18.06997436573494588029145420382

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