↑

Properties

Label 2-864-216.139-c0-0-0
Degree $2$
Conductor $864$
Sign $-0.116 + 0.993i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)9-s + (−0.266 − 1.50i)11-s + (0.939 − 1.62i)17-s + (0.173 + 0.300i)19-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)27-s + (−1.43 + 0.524i)33-s + (−0.326 + 0.118i)41-s + (0.326 + 1.85i)43-s + (0.173 − 0.984i)49-s + (−1.76 − 0.642i)51-s + (0.266 − 0.223i)57-s + (−0.0603 + 0.342i)59-s + (1.43 − 0.524i)67-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)3-s + (−0.939 + 0.342i)9-s + (−0.266 − 1.50i)11-s + (0.939 − 1.62i)17-s + (0.173 + 0.300i)19-s + (−0.939 − 0.342i)25-s + (0.5 + 0.866i)27-s + (−1.43 + 0.524i)33-s + (−0.326 + 0.118i)41-s + (0.326 + 1.85i)43-s + (0.173 − 0.984i)49-s + (−1.76 − 0.642i)51-s + (0.266 − 0.223i)57-s + (−0.0603 + 0.342i)59-s + (1.43 − 0.524i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.116 + 0.993i$
Analytic conductor: \(0.431192\)
Root analytic conductor: \(0.656652\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :0),\ -0.116 + 0.993i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8604669266\)
\(L(\frac12)\) \(\approx\) \(0.8604669266\)
\(L(1)\) 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 \( 1 + (0.173 + 0.984i)T \)
good5 \( 1 + (0.939 + 0.342i)T^{2} \)
7 \( 1 + (-0.173 + 0.984i)T^{2} \)
11 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (-0.173 - 0.984i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \)
43 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
47 \( 1 + (-0.173 + 0.984i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.0603 - 0.342i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.43 + 0.524i)T + (0.766 - 0.642i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.766 - 0.642i)T^{2} \)
83 \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.16120082854218140931947120499, −9.221692111433428009375311054032, −8.197001831674674982033674726616, −7.71482166484025105733971272805, −6.68623866776328465730773752713, −5.82120811786269020488655950824, −5.12943223799191162585901759655, −3.47266272589055496995336154381, −2.54837736362177468293033411053, −0.923867651047348099398639848761, 2.01109970149798648382494256556, 3.48038191394335792638921780288, 4.29690578283899426179963838290, 5.25725246553453368970832890388, 6.05937441505121674030918278128, 7.24873218296335389305051990575, 8.132563934465243470008556657694, 9.091584351918052383365857170490, 9.958519668671083339393124631448, 10.33681471956565338999840744755

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