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Properties

Label 2-18-9.4-c1-0-0
Degree $2$
Conductor $18$
Sign $0.939 + 0.342i$
Analytic cond. $0.143730$
Root an. cond. $0.379118$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 0.866i)2-s + (−1.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (1.5 + 0.866i)6-s + (−1 − 1.73i)7-s + 0.999·8-s + (1.5 − 2.59i)9-s + (1.5 + 2.59i)11-s − 1.73i·12-s + (−1 + 1.73i)13-s + (−0.999 + 1.73i)14-s + (−0.5 − 0.866i)16-s − 3·17-s − 3·18-s − 19-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.866 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.612 + 0.353i)6-s + (−0.377 − 0.654i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.452 + 0.783i)11-s − 0.499i·12-s + (−0.277 + 0.480i)13-s + (−0.267 + 0.462i)14-s + (−0.125 − 0.216i)16-s − 0.727·17-s − 0.707·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(18\)    =    \(2 \cdot 3^{2}\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(0.143730\)
Root analytic conductor: \(0.379118\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{18} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 18,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.410344 - 0.0723547i\)
\(L(\frac12)\) \(\approx\) \(0.410344 - 0.0723547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (1.5 - 0.866i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 3T + 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3 - 5.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.5 - 4.33i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 11T + 73T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (2.5 + 4.33i)T + (-48.5 + 84.0i)T^{2} \)
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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

−18.73425098025001598515848722623, −17.34259594350077650929836362354, −16.65829432309937211505679158926, −15.04331307163907501027393190909, −13.09153056920829555057476441677, −11.75914826730233688681995342362, −10.51435608625116348572044517930, −9.327762105443371388541282310587, −6.85650392855044452343093046473, −4.36179441556210697223984606647, 5.52147912858687978615646302053, 6.92583624002767656428145584125, 8.790108040571259251135981778049, 10.66744030764575546769971244074, 12.18420205922030737509979676105, 13.60256088276205419926735569181, 15.36554110674836471808657271021, 16.50555445570130060214767610978, 17.58588719176622164211034015698, 18.65330191638767616820202141858

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