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Properties

Label 2-288-9.7-c1-0-1
Degree $2$
Conductor $288$
Sign $0.450 - 0.892i$
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  + (−0.637 − 1.61i)3-s + (1.68 + 2.92i)5-s + (−2.35 + 4.07i)7-s + (−2.18 + 2.05i)9-s + (−0.437 + 0.758i)11-s + (0.686 + 1.18i)13-s + (3.62 − 4.57i)15-s − 2.37·17-s + 5.57·19-s + (8.05 + 1.18i)21-s + (−2.35 − 4.07i)23-s + (−3.18 + 5.51i)25-s + (4.70 + 2.20i)27-s + (2.68 − 4.65i)29-s + (3.22 + 5.58i)31-s + ⋯
L(s)  = 1  + (−0.368 − 0.929i)3-s + (0.754 + 1.30i)5-s + (−0.888 + 1.53i)7-s + (−0.728 + 0.684i)9-s + (−0.131 + 0.228i)11-s + (0.190 + 0.329i)13-s + (0.936 − 1.18i)15-s − 0.575·17-s + 1.27·19-s + (1.75 + 0.259i)21-s + (−0.490 − 0.849i)23-s + (−0.637 + 1.10i)25-s + (0.905 + 0.425i)27-s + (0.498 − 0.863i)29-s + (0.579 + 1.00i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.450 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.450 - 0.892i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.450 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.899998 + 0.553914i\)
\(L(\frac12)\) \(\approx\) \(0.899998 + 0.553914i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.637 + 1.61i)T \)
good5 \( 1 + (-1.68 - 2.92i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (2.35 - 4.07i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (0.437 - 0.758i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.686 - 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 2.37T + 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 + (2.35 + 4.07i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.68 + 4.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-3.22 - 5.58i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.437 + 0.758i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.35 - 4.07i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (4.26 + 7.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 + 1.83i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.26 - 7.38i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.40T + 71T^{2} \)
73 \( 1 - 10.3T + 73T^{2} \)
79 \( 1 + (3.22 - 5.58i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.47 + 2.55i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (4.5 - 7.79i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.01655174523271428784202014192, −11.20104438253484637502836080132, −10.09767422587562927277019333744, −9.233423764321983281558675211584, −8.022837797734694856056413815867, −6.60509565619710953764130840044, −6.38594382169025993340013511890, −5.32192992182478983576924522287, −2.95163567511660650148391584590, −2.23110178459491883231534488872, 0.844747079800542815861732081025, 3.40274227448249267615359237043, 4.48147036015176642210373706610, 5.45513769581762674236826534238, 6.47263953694936221403749624326, 7.895801209949622489886278484385, 9.198191997141157078962445602492, 9.738982416177387233802943556580, 10.47834179185199976473151149605, 11.52858417100605802139663582416

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