↑

Properties

Label 2-864-24.5-c2-0-9
Degree $2$
Conductor $864$
Sign $-0.0610 - 0.998i$
Analytic cond. $23.5422$
Root an. cond. $4.85204$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5.18·5-s − 3.67·7-s − 12.8·11-s − 8.72i·13-s + 18.6i·17-s + 30.7i·19-s + 20.7i·23-s + 1.88·25-s + 27.8·29-s + 16.6·31-s − 19.0·35-s + 60.5i·37-s − 28.2i·41-s + 51.8i·43-s − 69.2i·47-s + ⋯
L(s)  = 1  + 1.03·5-s − 0.524·7-s − 1.17·11-s − 0.670i·13-s + 1.09i·17-s + 1.61i·19-s + 0.903i·23-s + 0.0752·25-s + 0.961·29-s + 0.536·31-s − 0.544·35-s + 1.63i·37-s − 0.690i·41-s + 1.20i·43-s − 1.47i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0610 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0610 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $-0.0610 - 0.998i$
Analytic conductor: \(23.5422\)
Root analytic conductor: \(4.85204\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1),\ -0.0610 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.445093658\)
\(L(\frac12)\) \(\approx\) \(1.445093658\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 - 5.18T + 25T^{2} \)
7 \( 1 + 3.67T + 49T^{2} \)
11 \( 1 + 12.8T + 121T^{2} \)
13 \( 1 + 8.72iT - 169T^{2} \)
17 \( 1 - 18.6iT - 289T^{2} \)
19 \( 1 - 30.7iT - 361T^{2} \)
23 \( 1 - 20.7iT - 529T^{2} \)
29 \( 1 - 27.8T + 841T^{2} \)
31 \( 1 - 16.6T + 961T^{2} \)
37 \( 1 - 60.5iT - 1.36e3T^{2} \)
41 \( 1 + 28.2iT - 1.68e3T^{2} \)
43 \( 1 - 51.8iT - 1.84e3T^{2} \)
47 \( 1 + 69.2iT - 2.20e3T^{2} \)
53 \( 1 - 28.5T + 2.80e3T^{2} \)
59 \( 1 + 61.1T + 3.48e3T^{2} \)
61 \( 1 - 105. iT - 3.72e3T^{2} \)
67 \( 1 - 16.9iT - 4.48e3T^{2} \)
71 \( 1 + 26.1iT - 5.04e3T^{2} \)
73 \( 1 - 102.T + 5.32e3T^{2} \)
79 \( 1 - 92.2T + 6.24e3T^{2} \)
83 \( 1 - 36.4T + 6.88e3T^{2} \)
89 \( 1 - 73.5iT - 7.92e3T^{2} \)
97 \( 1 + 186.T + 9.40e3T^{2} \)
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

−10.18182797711918749541367077585, −9.631660199035944536785605602483, −8.350110245749744414161470296881, −7.84699411142720216242596107497, −6.50174436642805631276709298517, −5.84701211417451726763579078286, −5.11710914672064211500351778762, −3.67625677001262639122336639570, −2.64583160049567146988749853176, −1.44682105531776335658364388308, 0.46073639310339858992014882095, 2.25129495191437023476778053058, 2.91324643483325288972485970311, 4.56314251864200237956989745782, 5.29070398382014289156030202701, 6.35412725066336915492911605544, 6.99322430925687807828084339634, 8.083903281499387078422230945460, 9.223037154573640527786136747678, 9.572201225094807353414443118628

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