Properties

Degree $2$
Conductor $288$
Sign $0.997 + 0.0741i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.108 − 1.99i)2-s + (−3.97 − 0.432i)4-s + (2.81 + 1.16i)5-s + (−6.23 + 6.23i)7-s + (−1.29 + 7.89i)8-s + (2.63 − 5.49i)10-s + (8.06 + 3.33i)11-s + (13.3 − 5.51i)13-s + (11.7 + 13.1i)14-s + (15.6 + 3.43i)16-s + 4.56i·17-s + (13.4 + 32.4i)19-s + (−10.6 − 5.85i)20-s + (7.54 − 15.7i)22-s + (6.75 + 6.75i)23-s + ⋯
L(s)  = 1  + (0.0540 − 0.998i)2-s + (−0.994 − 0.108i)4-s + (0.563 + 0.233i)5-s + (−0.890 + 0.890i)7-s + (−0.161 + 0.986i)8-s + (0.263 − 0.549i)10-s + (0.732 + 0.303i)11-s + (1.02 − 0.424i)13-s + (0.841 + 0.937i)14-s + (0.976 + 0.214i)16-s + 0.268i·17-s + (0.707 + 1.70i)19-s + (−0.534 − 0.292i)20-s + (0.342 − 0.715i)22-s + (0.293 + 0.293i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0741i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0741i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.997 + 0.0741i$
Motivic weight: \(2\)
Character: $\chi_{288} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.997 + 0.0741i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.49301 - 0.0554608i\)
\(L(\frac12)\) \(\approx\) \(1.49301 - 0.0554608i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.108 + 1.99i)T \)
3 \( 1 \)
good5 \( 1 + (-2.81 - 1.16i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (6.23 - 6.23i)T - 49iT^{2} \)
11 \( 1 + (-8.06 - 3.33i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (-13.3 + 5.51i)T + (119. - 119. i)T^{2} \)
17 \( 1 - 4.56iT - 289T^{2} \)
19 \( 1 + (-13.4 - 32.4i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (-6.75 - 6.75i)T + 529iT^{2} \)
29 \( 1 + (-0.266 - 0.643i)T + (-594. + 594. i)T^{2} \)
31 \( 1 + 0.326iT - 961T^{2} \)
37 \( 1 + (-31.5 - 13.0i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (15.7 - 15.7i)T - 1.68e3iT^{2} \)
43 \( 1 + (-4.83 - 2.00i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 - 49.7T + 2.20e3T^{2} \)
53 \( 1 + (4.45 - 10.7i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (13.1 - 31.6i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (35.4 + 85.4i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (41.3 - 17.1i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (37.6 - 37.6i)T - 5.04e3iT^{2} \)
73 \( 1 + (52.2 - 52.2i)T - 5.32e3iT^{2} \)
79 \( 1 - 26.9T + 6.24e3T^{2} \)
83 \( 1 + (10.6 + 25.6i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-103. - 103. i)T + 7.92e3iT^{2} \)
97 \( 1 - 77.9T + 9.40e3T^{2} \)
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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

−11.72904001849628535142829596566, −10.54927739879581640436804094831, −9.767199444663358250546486281509, −9.101894148872037566743112067532, −8.028309306765297988560277760381, −6.23170622312628836658060254981, −5.64443544342000230354063406595, −3.94556599509466488612195687569, −2.93092386647631057371226901178, −1.51038417364906224770901005559, 0.829972253640935676595418798848, 3.39881231041580404069890737784, 4.50367380322955154096601181974, 5.83906330186800323857650888500, 6.66357413696312377723939739794, 7.43917541678378929210005001632, 8.978328059862167310961480009461, 9.290643147201677736351255339323, 10.44629890880778967232533003877, 11.62899059311363750783333777214

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