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Properties

Label 2-288-72.59-c1-0-4
Degree $2$
Conductor $288$
Sign $0.866 - 0.498i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
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  + (1.71 − 0.231i)3-s + (−1.74 + 3.01i)5-s + (1.80 − 1.04i)7-s + (2.89 − 0.795i)9-s + (0.116 − 0.0675i)11-s + (2.63 + 1.52i)13-s + (−2.29 + 5.58i)15-s + 4.19i·17-s − 0.919·19-s + (2.86 − 2.21i)21-s + (0.689 − 1.19i)23-s + (−3.57 − 6.19i)25-s + (4.78 − 2.03i)27-s + (−4.24 − 7.34i)29-s + (−4.39 − 2.53i)31-s + ⋯
L(s)  = 1  + (0.990 − 0.133i)3-s + (−0.779 + 1.35i)5-s + (0.683 − 0.394i)7-s + (0.964 − 0.265i)9-s + (0.0352 − 0.0203i)11-s + (0.731 + 0.422i)13-s + (−0.591 + 1.44i)15-s + 1.01i·17-s − 0.210·19-s + (0.624 − 0.482i)21-s + (0.143 − 0.249i)23-s + (−0.715 − 1.23i)25-s + (0.919 − 0.392i)27-s + (−0.787 − 1.36i)29-s + (−0.790 − 0.456i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.866 - 0.498i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.866 - 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.62747 + 0.434412i\)
\(L(\frac12)\) \(\approx\) \(1.62747 + 0.434412i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 + 0.231i)T \)
good5 \( 1 + (1.74 - 3.01i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.80 + 1.04i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.116 + 0.0675i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.63 - 1.52i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.19iT - 17T^{2} \)
19 \( 1 + 0.919T + 19T^{2} \)
23 \( 1 + (-0.689 + 1.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.24 + 7.34i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.39 + 2.53i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.61iT - 37T^{2} \)
41 \( 1 + (-1.79 - 1.03i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.41 + 9.37i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.205 + 0.356i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 0.968T + 53T^{2} \)
59 \( 1 + (3.88 + 2.24i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.44 + 4.29i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.15 - 5.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.9T + 71T^{2} \)
73 \( 1 + 4.06T + 73T^{2} \)
79 \( 1 + (-10.8 + 6.27i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.23 - 3.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.35iT - 89T^{2} \)
97 \( 1 + (0.477 + 0.826i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.68641336571356233205195032376, −10.93407034941293253870642892102, −10.15746765426202611613594546480, −8.835862773286322231893862731156, −7.921356080985426767355671510526, −7.26783788674463247335374907347, −6.23752710215516671034734301884, −4.19126506004768187893331332376, −3.51092214629291842297694457049, −2.03861373431914443137651071921, 1.47077873464823914621433879403, 3.29669173497405712967341415373, 4.49973344559955943994305371136, 5.33156311084971040702763535342, 7.23200445321509041186366343157, 8.161237265467829894239958581021, 8.744739882305154388744675320600, 9.462880869611443217675141453840, 10.88419976222293711802287707822, 11.83503940120176517157235542971

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