↑

Properties

Label 2-864-24.11-c3-0-27
Degree $2$
Conductor $864$
Sign $0.829 - 0.558i$
Analytic cond. $50.9776$
Root an. cond. $7.13986$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 21.0·5-s + 25.8i·7-s − 46.9i·11-s + 55.4i·13-s − 64.8i·17-s + 51.0·19-s + 60.4·23-s + 316.·25-s + 90.5·29-s − 8.35i·31-s + 542. i·35-s + 228. i·37-s − 239. i·41-s + 192.·43-s + 38.1·47-s + ⋯
L(s)  = 1  + 1.88·5-s + 1.39i·7-s − 1.28i·11-s + 1.18i·13-s − 0.924i·17-s + 0.616·19-s + 0.548·23-s + 2.53·25-s + 0.579·29-s − 0.0484i·31-s + 2.62i·35-s + 1.01i·37-s − 0.913i·41-s + 0.684·43-s + 0.118·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.829 - 0.558i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.829 - 0.558i$
Analytic conductor: \(50.9776\)
Root analytic conductor: \(7.13986\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :3/2),\ 0.829 - 0.558i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.213327744\)
\(L(\frac12)\) \(\approx\) \(3.213327744\)
\(L(\frac{5}{2})\) 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 \( 1 - 21.0T + 125T^{2} \)
7 \( 1 - 25.8iT - 343T^{2} \)
11 \( 1 + 46.9iT - 1.33e3T^{2} \)
13 \( 1 - 55.4iT - 2.19e3T^{2} \)
17 \( 1 + 64.8iT - 4.91e3T^{2} \)
19 \( 1 - 51.0T + 6.85e3T^{2} \)
23 \( 1 - 60.4T + 1.21e4T^{2} \)
29 \( 1 - 90.5T + 2.43e4T^{2} \)
31 \( 1 + 8.35iT - 2.97e4T^{2} \)
37 \( 1 - 228. iT - 5.06e4T^{2} \)
41 \( 1 + 239. iT - 6.89e4T^{2} \)
43 \( 1 - 192.T + 7.95e4T^{2} \)
47 \( 1 - 38.1T + 1.03e5T^{2} \)
53 \( 1 + 6.69T + 1.48e5T^{2} \)
59 \( 1 + 5.40iT - 2.05e5T^{2} \)
61 \( 1 - 847. iT - 2.26e5T^{2} \)
67 \( 1 + 992.T + 3.00e5T^{2} \)
71 \( 1 - 494.T + 3.57e5T^{2} \)
73 \( 1 + 395.T + 3.89e5T^{2} \)
79 \( 1 - 42.1iT - 4.93e5T^{2} \)
83 \( 1 + 328. iT - 5.71e5T^{2} \)
89 \( 1 - 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 - 991.T + 9.12e5T^{2} \)
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

−9.623318241983926944355754609043, −9.051799706489444900165150473602, −8.630688516586291570440254220619, −7.04609818993173496869615578509, −6.12785118477627655093740000984, −5.65139018596537980809512071511, −4.83947858079614957541278649017, −3.00450934626298800388682072120, −2.33667111273676322985064888882, −1.19447957604360650670534178026, 0.954215884110134188102371407414, 1.88753243212341342145151051679, 3.08998158436784645533027477802, 4.44498094609080631953067442971, 5.32727140315045639964898386407, 6.22482167591129227434354319982, 7.06470372949295171822954255479, 7.86339456139045867675770824126, 9.127791219608657816265655593637, 9.897579231086577298347239921233

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