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Properties

Label 2-18e2-4.3-c2-0-20
Degree $2$
Conductor $324$
Sign $-0.0581 - 0.998i$
Analytic cond. $8.82836$
Root an. cond. $2.97125$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.45 + 1.37i)2-s + (0.232 + 3.99i)4-s + 4.37·5-s − 2.13i·7-s + (−5.14 + 6.12i)8-s + (6.36 + 6.00i)10-s + 12.6i·11-s + 9.26·13-s + (2.93 − 3.10i)14-s + (−15.8 + 1.85i)16-s + 14.6·17-s + 34.5i·19-s + (1.01 + 17.4i)20-s + (−17.3 + 18.3i)22-s − 40.7i·23-s + ⋯
L(s)  = 1  + (0.727 + 0.686i)2-s + (0.0581 + 0.998i)4-s + 0.874·5-s − 0.305i·7-s + (−0.642 + 0.766i)8-s + (0.636 + 0.600i)10-s + 1.14i·11-s + 0.712·13-s + (0.209 − 0.221i)14-s + (−0.993 + 0.116i)16-s + 0.861·17-s + 1.81i·19-s + (0.0508 + 0.873i)20-s + (−0.787 + 0.834i)22-s − 1.77i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0581 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.0581 - 0.998i$
Analytic conductor: \(8.82836\)
Root analytic conductor: \(2.97125\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :1),\ -0.0581 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.88824 + 2.00145i\)
\(L(\frac12)\) \(\approx\) \(1.88824 + 2.00145i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.45 - 1.37i)T \)
3 \( 1 \)
good5 \( 1 - 4.37T + 25T^{2} \)
7 \( 1 + 2.13iT - 49T^{2} \)
11 \( 1 - 12.6iT - 121T^{2} \)
13 \( 1 - 9.26T + 169T^{2} \)
17 \( 1 - 14.6T + 289T^{2} \)
19 \( 1 - 34.5iT - 361T^{2} \)
23 \( 1 + 40.7iT - 529T^{2} \)
29 \( 1 - 19.0T + 841T^{2} \)
31 \( 1 + 0.993iT - 961T^{2} \)
37 \( 1 + 66.4T + 1.36e3T^{2} \)
41 \( 1 - 25.8T + 1.68e3T^{2} \)
43 \( 1 - 42.1iT - 1.84e3T^{2} \)
47 \( 1 + 34.7iT - 2.20e3T^{2} \)
53 \( 1 + 12.2T + 2.80e3T^{2} \)
59 \( 1 + 56.7iT - 3.48e3T^{2} \)
61 \( 1 - 73.7T + 3.72e3T^{2} \)
67 \( 1 + 95.1iT - 4.48e3T^{2} \)
71 \( 1 + 75.5iT - 5.04e3T^{2} \)
73 \( 1 + 56.7T + 5.32e3T^{2} \)
79 \( 1 + 74.5iT - 6.24e3T^{2} \)
83 \( 1 + 65.4iT - 6.88e3T^{2} \)
89 \( 1 - 150.T + 7.92e3T^{2} \)
97 \( 1 + 112.T + 9.40e3T^{2} \)
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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

−12.12090024968682081935254245669, −10.57297707158069448459228521834, −9.854785030581807334780179564798, −8.592408174335938519109175426896, −7.68220310011742659946544154724, −6.57271440882886449156759966368, −5.81899948362685481559376816154, −4.70625366319570913444676343566, −3.55468903140619532566820123521, −1.95920033568331579770233306318, 1.15731195499420197718237707953, 2.64612403004092162151166691107, 3.72201226965349808853110949283, 5.34339344972764106516134379000, 5.79907859752791450625678988083, 6.98469966510434410691474120148, 8.673146225890591678153792421864, 9.428956038571983120438217875899, 10.41297887800152199954822932100, 11.25783316485745392430178350290

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