Properties

Label 2-864-216.67-c0-0-0
Degree $2$
Conductor $864$
Sign $0.918 - 0.396i$
Analytic cond. $0.431192$
Root an. cond. $0.656652$
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  + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)9-s + (1.43 − 1.20i)11-s + (−0.173 + 0.300i)17-s + (0.766 + 1.32i)19-s + (0.173 + 0.984i)25-s + (0.500 + 0.866i)27-s + (−0.326 + 1.85i)33-s + (0.266 − 1.50i)41-s + (−0.266 + 0.223i)43-s + (0.766 + 0.642i)49-s + (−0.0603 − 0.342i)51-s + (−1.43 − 0.524i)57-s + (−1.17 − 0.984i)59-s + (0.326 − 1.85i)67-s + ⋯
L(s)  = 1  + (−0.766 + 0.642i)3-s + (0.173 − 0.984i)9-s + (1.43 − 1.20i)11-s + (−0.173 + 0.300i)17-s + (0.766 + 1.32i)19-s + (0.173 + 0.984i)25-s + (0.500 + 0.866i)27-s + (−0.326 + 1.85i)33-s + (0.266 − 1.50i)41-s + (−0.266 + 0.223i)43-s + (0.766 + 0.642i)49-s + (−0.0603 − 0.342i)51-s + (−1.43 − 0.524i)57-s + (−1.17 − 0.984i)59-s + (0.326 − 1.85i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.918 - 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.918 - 0.396i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :0),\ 0.918 - 0.396i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8430595438\)
\(L(\frac12)\) \(\approx\) \(0.8430595438\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.766 - 0.642i)T \)
good5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (-0.766 - 0.642i)T^{2} \)
11 \( 1 + (-1.43 + 1.20i)T + (0.173 - 0.984i)T^{2} \)
13 \( 1 + (0.939 + 0.342i)T^{2} \)
17 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (-0.766 + 0.642i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (0.266 - 0.223i)T + (0.173 - 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
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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

−10.60217938060682259833517069392, −9.483789513685366094211440820451, −9.048590610722006175178668135896, −7.934542031871418280105880249076, −6.72904052626321690493976237270, −6.00479164894248623330059057604, −5.27353005254842872102181965405, −3.97024591520817943054651417412, −3.41635591593603092218720761330, −1.30184874323710865942771920751, 1.27439171901337151242175599893, 2.58004789456972379620689128136, 4.25728877682293028888567227115, 4.96465952706016470716110972001, 6.15767982291312986881143777178, 6.90469640220417199107341311042, 7.43566986774318714601768681980, 8.665370753904041975799285980307, 9.538086466294714446199218127048, 10.32603388636591074661355910632

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