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Properties

Label 2-18e2-36.7-c0-0-1
Degree $2$
Conductor $324$
Sign $-0.173 + 0.984i$
Analytic cond. $0.161697$
Root an. cond. $0.402115$
Motivic weight $0$
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 + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·8-s − 0.999·10-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s + (−0.499 + 0.866i)20-s + 0.999·26-s + (−0.5 + 0.866i)29-s + (0.499 + 0.866i)32-s + (0.5 − 0.866i)34-s − 37-s + (0.499 + 0.866i)40-s + (1 + 1.73i)41-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s − 0.999·8-s − 0.999·10-s + (0.5 + 0.866i)13-s + (−0.5 + 0.866i)16-s + 17-s + (−0.499 + 0.866i)20-s + 0.999·26-s + (−0.5 + 0.866i)29-s + (0.499 + 0.866i)32-s + (0.5 − 0.866i)34-s − 37-s + (0.499 + 0.866i)40-s + (1 + 1.73i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(324\)    =    \(2^{2} \cdot 3^{4}\)
Sign: $-0.173 + 0.984i$
Analytic conductor: \(0.161697\)
Root analytic conductor: \(0.402115\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{324} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 324,\ (\ :0),\ -0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8880677672\)
\(L(\frac12)\) \(\approx\) \(0.8880677672\)
\(L(1)\) 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 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + 2T + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T + T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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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

−11.68877308008808384256828906453, −10.90606169435844360388427052995, −9.766510012575916346152496652124, −8.970394260126961992625844473835, −8.024495598177834488461869046055, −6.50657644292250447716540130405, −5.28485827507888118730371888742, −4.37811389863428559604401033731, −3.30140811139359753917009823183, −1.48700136974751797992693691160, 3.01902277136168782070601813476, 3.89964760933651547210761597633, 5.34915903434840049737878130599, 6.25052909631282171134968692780, 7.37799867925631147212916497772, 7.927584350201017393692245921855, 9.083789136023237763731663894118, 10.31085536518895238848043204798, 11.27429802528465711480310233111, 12.24675039251961105686052911443

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