Properties

Degree $2$
Conductor $288$
Sign $-0.507 + 0.861i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 0.488i)2-s + (3.52 − 1.89i)4-s + (1.85 + 4.48i)5-s + (−5.27 − 5.27i)7-s + (−5.90 + 5.40i)8-s + (−5.79 − 7.79i)10-s + (−6.20 − 14.9i)11-s + (−4.22 + 10.2i)13-s + (12.8 + 7.65i)14-s + (8.80 − 13.3i)16-s + 2.84i·17-s + (−12.4 − 5.14i)19-s + (15.0 + 12.2i)20-s + (19.3 + 25.9i)22-s + (1.43 − 1.43i)23-s + ⋯
L(s)  = 1  + (−0.969 + 0.244i)2-s + (0.880 − 0.474i)4-s + (0.371 + 0.897i)5-s + (−0.753 − 0.753i)7-s + (−0.737 + 0.675i)8-s + (−0.579 − 0.779i)10-s + (−0.563 − 1.36i)11-s + (−0.325 + 0.784i)13-s + (0.915 + 0.546i)14-s + (0.550 − 0.834i)16-s + 0.167i·17-s + (−0.654 − 0.270i)19-s + (0.752 + 0.613i)20-s + (0.879 + 1.18i)22-s + (0.0625 − 0.0625i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.507 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.507 + 0.861i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.507 + 0.861i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.174116 - 0.304595i\)
\(L(\frac12)\) \(\approx\) \(0.174116 - 0.304595i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.93 - 0.488i)T \)
3 \( 1 \)
good5 \( 1 + (-1.85 - 4.48i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (5.27 + 5.27i)T + 49iT^{2} \)
11 \( 1 + (6.20 + 14.9i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (4.22 - 10.2i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 2.84iT - 289T^{2} \)
19 \( 1 + (12.4 + 5.14i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-1.43 + 1.43i)T - 529iT^{2} \)
29 \( 1 + (36.9 + 15.3i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 4.73iT - 961T^{2} \)
37 \( 1 + (6.68 + 16.1i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (40.4 + 40.4i)T + 1.68e3iT^{2} \)
43 \( 1 + (24.5 + 59.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 16.5T + 2.20e3T^{2} \)
53 \( 1 + (-46.9 + 19.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (50.0 - 20.7i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (54.3 + 22.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (25.5 - 61.5i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (7.12 + 7.12i)T + 5.04e3iT^{2} \)
73 \( 1 + (-55.3 - 55.3i)T + 5.32e3iT^{2} \)
79 \( 1 - 11.0T + 6.24e3T^{2} \)
83 \( 1 + (-29.9 - 12.4i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (16.7 - 16.7i)T - 7.92e3iT^{2} \)
97 \( 1 + 67.8T + 9.40e3T^{2} \)
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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

−10.78225373606256820796161252742, −10.46925768868150262615741966916, −9.416879385720621610440861060447, −8.468528651191402644408431244721, −7.24518782324112579888391832429, −6.60385753762402974883660334901, −5.65077734428282158797929117152, −3.57796976312955666477605132833, −2.30174462114724317667755173702, −0.21994034022113161976100785387, 1.74669789698229543871384515255, 3.02129103587857336813359256154, 4.89199022477922275360369059277, 6.05267708871454801050093211577, 7.26462402429323393239057260712, 8.258176098406360544024213392067, 9.263675765830254907482614561928, 9.775348993727310508934642399292, 10.69912954694287593237295815999, 12.04357189843312043823817429550

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