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Properties

Label 2-864-36.11-c1-0-3
Degree $2$
Conductor $864$
Sign $0.665 + 0.745i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.40 + 1.96i)5-s + (−0.961 − 0.555i)7-s + (1.63 − 2.82i)11-s + (0.124 + 0.216i)13-s − 5.86i·17-s + 2.19i·19-s + (2.79 + 4.83i)23-s + (5.24 − 9.09i)25-s + (−2.35 − 1.36i)29-s + (8.96 − 5.17i)31-s + 4.36·35-s − 0.333·37-s + (5.28 − 3.05i)41-s + (8.50 + 4.91i)43-s + (4.70 − 8.15i)47-s + ⋯
L(s)  = 1  + (−1.52 + 0.880i)5-s + (−0.363 − 0.209i)7-s + (0.492 − 0.852i)11-s + (0.0346 + 0.0600i)13-s − 1.42i·17-s + 0.504i·19-s + (0.582 + 1.00i)23-s + (1.04 − 1.81i)25-s + (−0.437 − 0.252i)29-s + (1.61 − 0.930i)31-s + 0.738·35-s − 0.0547·37-s + (0.825 − 0.476i)41-s + (1.29 + 0.749i)43-s + (0.686 − 1.18i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.745i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.665 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(864\)    =    \(2^{5} \cdot 3^{3}\)
Sign: $0.665 + 0.745i$
Analytic conductor: \(6.89907\)
Root analytic conductor: \(2.62660\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{864} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 864,\ (\ :1/2),\ 0.665 + 0.745i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.867855 - 0.388586i\)
\(L(\frac12)\) \(\approx\) \(0.867855 - 0.388586i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (3.40 - 1.96i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.961 + 0.555i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.63 + 2.82i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.124 - 0.216i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.86iT - 17T^{2} \)
19 \( 1 - 2.19iT - 19T^{2} \)
23 \( 1 + (-2.79 - 4.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (2.35 + 1.36i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-8.96 + 5.17i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (-5.28 + 3.05i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-8.50 - 4.91i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.70 + 8.15i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4.75iT - 53T^{2} \)
59 \( 1 + (3.26 + 5.65i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.07 + 1.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.501 - 0.289i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 3.26T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + (7.67 + 4.42i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.34 - 4.05i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 + (-0.916 + 1.58i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.10766747035929979537817854605, −9.213119194647548708277840307424, −8.188956481942846506137467599055, −7.45253990771435622980015144930, −6.82336751165519668712006082383, −5.79712640889674251255563868851, −4.40501625787711493214684535121, −3.57417231934171119958114318252, −2.81363383014867272299183005087, −0.56696938341494541877386683937, 1.13759230099744394913054573584, 2.93201078668759696437388324793, 4.23901092349521762732918531412, 4.53465301261639225034551350548, 5.94712676674339943047870950968, 7.00097270423045196371052788745, 7.78905588463433719258734991850, 8.658250435699490743819370507982, 9.149160834271463776980953465207, 10.37276814954999839369724147951

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