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: $\chi_{288} (235, \cdot )$
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

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

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