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Properties

Label 2-288-36.11-c1-0-1
Degree $2$
Conductor $288$
Sign $-0.687 - 0.726i$
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  + (−0.324 + 1.70i)3-s + (−1.68 + 0.971i)5-s + (2.61 + 1.50i)7-s + (−2.78 − 1.10i)9-s + (−2.20 + 3.81i)11-s + (−2.65 − 4.60i)13-s + (−1.10 − 3.17i)15-s + 4.16i·17-s + 4.66i·19-s + (−3.41 + 3.95i)21-s + (−1.30 − 2.26i)23-s + (−0.613 + 1.06i)25-s + (2.78 − 4.38i)27-s + (1.13 + 0.655i)29-s + (0.0648 − 0.0374i)31-s + ⋯
L(s)  = 1  + (−0.187 + 0.982i)3-s + (−0.752 + 0.434i)5-s + (0.988 + 0.570i)7-s + (−0.929 − 0.367i)9-s + (−0.664 + 1.15i)11-s + (−0.737 − 1.27i)13-s + (−0.285 − 0.820i)15-s + 1.00i·17-s + 1.06i·19-s + (−0.745 + 0.863i)21-s + (−0.272 − 0.471i)23-s + (−0.122 + 0.212i)25-s + (0.535 − 0.844i)27-s + (0.210 + 0.121i)29-s + (0.0116 − 0.00672i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.687 - 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.687 - 0.726i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.687 - 0.726i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.360318 + 0.837110i\)
\(L(\frac12)\) \(\approx\) \(0.360318 + 0.837110i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.324 - 1.70i)T \)
good5 \( 1 + (1.68 - 0.971i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.61 - 1.50i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.20 - 3.81i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.65 + 4.60i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 4.16iT - 17T^{2} \)
19 \( 1 - 4.66iT - 19T^{2} \)
23 \( 1 + (1.30 + 2.26i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.13 - 0.655i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.0648 + 0.0374i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.43T + 37T^{2} \)
41 \( 1 + (-8.16 + 4.71i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.06 - 2.92i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.40 - 4.16i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 8.96iT - 53T^{2} \)
59 \( 1 + (-1.74 - 3.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6.32 + 10.9i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.38 - 1.37i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.910T + 71T^{2} \)
73 \( 1 - 6.86T + 73T^{2} \)
79 \( 1 + (-10.8 - 6.28i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.61 + 14.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.96iT - 89T^{2} \)
97 \( 1 + (9.03 - 15.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.12198074973919921118113899642, −11.01145461544295190891051817556, −10.44503142761309397848043120065, −9.516763429600991608871584906881, −8.097400084698294575638056063551, −7.72330991710846178542824689108, −5.92449832962760776152228578847, −4.97244223274308149295308022594, −3.98895213045744473096696853679, −2.54002781617852592579175310001, 0.70161363433534090150035767920, 2.50660897425202963427928947750, 4.34183142223396735495874958575, 5.31932629779480402070402557258, 6.75656985703428641336421239905, 7.67239712219149027572678111052, 8.255149562318860808133026972530, 9.379237796919567536664219475426, 11.10148567417500103285066499494, 11.40665828997420110182211557215

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