Properties

Label 2-864-216.155-c1-0-4
Degree $2$
Conductor $864$
Sign $0.158 - 0.987i$
Analytic cond. $6.89907$
Root an. cond. $2.62660$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.21 − 1.23i)3-s + (−0.0276 + 2.99i)9-s + (−1.66 − 0.293i)11-s + (−2.82 + 1.63i)17-s + (−4.00 + 6.93i)19-s + (4.69 − 1.71i)25-s + (3.72 − 3.62i)27-s + (1.66 + 2.40i)33-s + (−2.69 + 7.39i)41-s + (−2.13 + 12.1i)43-s + (1.21 + 6.89i)49-s + (5.45 + 1.48i)51-s + (13.4 − 3.52i)57-s + (−3.40 + 0.599i)59-s + (15.2 + 5.53i)67-s + ⋯
L(s)  = 1  + (−0.703 − 0.710i)3-s + (−0.00922 + 0.999i)9-s + (−0.501 − 0.0884i)11-s + (−0.685 + 0.395i)17-s + (−0.918 + 1.59i)19-s + (0.939 − 0.342i)25-s + (0.716 − 0.697i)27-s + (0.290 + 0.418i)33-s + (−0.420 + 1.15i)41-s + (−0.325 + 1.84i)43-s + (0.173 + 0.984i)49-s + (0.763 + 0.208i)51-s + (1.77 − 0.467i)57-s + (−0.442 + 0.0780i)59-s + (1.85 + 0.676i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.158 - 0.987i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.158 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.503162 + 0.428781i\)
\(L(\frac12)\) \(\approx\) \(0.503162 + 0.428781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.21 + 1.23i)T \)
good5 \( 1 + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (-1.21 - 6.89i)T^{2} \)
11 \( 1 + (1.66 + 0.293i)T + (10.3 + 3.76i)T^{2} \)
13 \( 1 + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (2.82 - 1.63i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.00 - 6.93i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.99 - 22.6i)T^{2} \)
29 \( 1 + (22.2 - 18.6i)T^{2} \)
31 \( 1 + (-5.38 + 30.5i)T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.69 - 7.39i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (2.13 - 12.1i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (8.16 + 46.2i)T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + (3.40 - 0.599i)T + (55.4 - 20.1i)T^{2} \)
61 \( 1 + (-10.5 - 60.0i)T^{2} \)
67 \( 1 + (-15.2 - 5.53i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.84 + 6.65i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (-5.81 - 15.9i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (11.0 + 6.38i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.22 - 18.2i)T + (-91.1 - 33.1i)T^{2} \)
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

−10.60238707107611850259283831745, −9.645468353995393664013738323368, −8.315510460988879148650815470614, −7.946269278357050946147352162256, −6.70586424635710125140066395175, −6.17989773465019080433676210876, −5.17396306324662476767537433383, −4.19157527619255992945865325572, −2.66393891762056122302969706177, −1.43413082899432733285236748385, 0.35509473853975949395123788390, 2.40873812781434473652747104929, 3.70303236448730302430179062586, 4.77834299870269886849809424195, 5.33476067369977228630927789551, 6.58090096568385114075258798666, 7.12352393316916464992163461333, 8.605466616194059214722500578101, 9.108440488477457511470219535165, 10.14184755471504777654065722977

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