Properties

Degree $2$
Conductor $288$
Sign $0.396 + 0.917i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.96 + 0.384i)2-s + (3.70 + 1.50i)4-s + (−2.95 − 7.13i)5-s + (−4.18 − 4.18i)7-s + (6.69 + 4.38i)8-s + (−3.05 − 15.1i)10-s + (−1.42 − 3.44i)11-s + (8.39 − 20.2i)13-s + (−6.60 − 9.82i)14-s + (11.4 + 11.1i)16-s − 1.73i·17-s + (14.2 + 5.90i)19-s + (−0.187 − 30.8i)20-s + (−1.47 − 7.30i)22-s + (−15.1 + 15.1i)23-s + ⋯
L(s)  = 1  + (0.981 + 0.192i)2-s + (0.926 + 0.377i)4-s + (−0.591 − 1.42i)5-s + (−0.597 − 0.597i)7-s + (0.836 + 0.547i)8-s + (−0.305 − 1.51i)10-s + (−0.129 − 0.313i)11-s + (0.646 − 1.55i)13-s + (−0.471 − 0.701i)14-s + (0.715 + 0.698i)16-s − 0.101i·17-s + (0.749 + 0.310i)19-s + (−0.00938 − 1.54i)20-s + (−0.0671 − 0.332i)22-s + (−0.658 + 0.658i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.396 + 0.917i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.396 + 0.917i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.08403 - 1.36971i\)
\(L(\frac12)\) \(\approx\) \(2.08403 - 1.36971i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.96 - 0.384i)T \)
3 \( 1 \)
good5 \( 1 + (2.95 + 7.13i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (4.18 + 4.18i)T + 49iT^{2} \)
11 \( 1 + (1.42 + 3.44i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-8.39 + 20.2i)T + (-119. - 119. i)T^{2} \)
17 \( 1 + 1.73iT - 289T^{2} \)
19 \( 1 + (-14.2 - 5.90i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (15.1 - 15.1i)T - 529iT^{2} \)
29 \( 1 + (-6.74 - 2.79i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + (-5.30 - 12.7i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-18.5 - 18.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (-31.0 - 75.0i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 16.2T + 2.20e3T^{2} \)
53 \( 1 + (-29.0 + 12.0i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (34.1 - 14.1i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (68.7 + 28.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-10.5 + 25.3i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-32.2 - 32.2i)T + 5.04e3iT^{2} \)
73 \( 1 + (28.5 + 28.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 22.4T + 6.24e3T^{2} \)
83 \( 1 + (-123. - 51.0i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (61.0 - 61.0i)T - 7.92e3iT^{2} \)
97 \( 1 + 69.9T + 9.40e3T^{2} \)
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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

−11.71272628217709538157956491453, −10.71295041244082075485291131660, −9.542120906664464428002653561988, −8.109375571558069576358348635708, −7.70735514334631626467703747510, −6.12778087014638861404494683488, −5.28247530428003192211884018832, −4.15354814606961324659061645505, −3.21116653376432513422896298872, −0.945026392773175438984498583036, 2.25379315679131047617118849934, 3.32680207101774591434151420492, 4.31399922084516186606733691334, 5.90155727800828082791091314886, 6.72540186991010467077168602139, 7.40724813380413422566519138271, 9.028803643148513072374290442736, 10.26166591252972699130549027265, 11.01904204467335649048162648471, 11.85122563670384309567680101584

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