Properties

Degree $2$
Conductor $288$
Sign $0.941 - 0.338i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.82 + 0.820i)2-s + (2.65 − 2.99i)4-s + (7.60 − 3.15i)5-s + (6.84 + 6.84i)7-s + (−2.39 + 7.63i)8-s + (−11.2 + 11.9i)10-s + (2.23 − 0.927i)11-s + (1.40 + 0.583i)13-s + (−18.0 − 6.86i)14-s + (−1.90 − 15.8i)16-s − 2.67i·17-s + (5.38 − 13.0i)19-s + (10.7 − 31.1i)20-s + (−3.32 + 3.52i)22-s + (−18.8 + 18.8i)23-s + ⋯
L(s)  = 1  + (−0.912 + 0.410i)2-s + (0.663 − 0.747i)4-s + (1.52 − 0.630i)5-s + (0.977 + 0.977i)7-s + (−0.298 + 0.954i)8-s + (−1.12 + 1.19i)10-s + (0.203 − 0.0842i)11-s + (0.108 + 0.0449i)13-s + (−1.29 − 0.490i)14-s + (−0.118 − 0.992i)16-s − 0.157i·17-s + (0.283 − 0.684i)19-s + (0.538 − 1.55i)20-s + (−0.151 + 0.160i)22-s + (−0.819 + 0.819i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.941 - 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.941 - 0.338i$
Motivic weight: \(2\)
Character: $\chi_{288} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.941 - 0.338i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.53534 + 0.267669i\)
\(L(\frac12)\) \(\approx\) \(1.53534 + 0.267669i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.82 - 0.820i)T \)
3 \( 1 \)
good5 \( 1 + (-7.60 + 3.15i)T + (17.6 - 17.6i)T^{2} \)
7 \( 1 + (-6.84 - 6.84i)T + 49iT^{2} \)
11 \( 1 + (-2.23 + 0.927i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-1.40 - 0.583i)T + (119. + 119. i)T^{2} \)
17 \( 1 + 2.67iT - 289T^{2} \)
19 \( 1 + (-5.38 + 13.0i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + (18.8 - 18.8i)T - 529iT^{2} \)
29 \( 1 + (-10.0 + 24.2i)T + (-594. - 594. i)T^{2} \)
31 \( 1 - 47.5iT - 961T^{2} \)
37 \( 1 + (28.2 - 11.7i)T + (968. - 968. i)T^{2} \)
41 \( 1 + (6.93 + 6.93i)T + 1.68e3iT^{2} \)
43 \( 1 + (-8.48 + 3.51i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 - 67.0T + 2.20e3T^{2} \)
53 \( 1 + (-10.5 - 25.3i)T + (-1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (27.9 + 67.4i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-31.5 + 76.2i)T + (-2.63e3 - 2.63e3i)T^{2} \)
67 \( 1 + (-90.1 - 37.3i)T + (3.17e3 + 3.17e3i)T^{2} \)
71 \( 1 + (1.98 + 1.98i)T + 5.04e3iT^{2} \)
73 \( 1 + (55.5 + 55.5i)T + 5.32e3iT^{2} \)
79 \( 1 - 10.9T + 6.24e3T^{2} \)
83 \( 1 + (34.1 - 82.5i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (-16.1 + 16.1i)T - 7.92e3iT^{2} \)
97 \( 1 + 62.6T + 9.40e3T^{2} \)
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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

−11.52091371254681781332349332208, −10.43951816951244107818212089666, −9.486316815719883371114907484208, −8.897976951959864083507076074063, −8.087058017930129979935374500069, −6.67269409672377889823834140861, −5.62772644470639937210113560920, −5.07364885186505991303334526639, −2.36242655441156403748097031296, −1.40242748514074230472030748860, 1.33726848360653434217385854282, 2.41110330711301742993185591231, 4.03086795266889472742376247371, 5.75341867377699704680631012409, 6.80787344616930432965846456991, 7.73581103562713297580038619964, 8.819784039894862634599629344691, 9.939128196993051024870121704796, 10.41435081882230324366319396998, 11.14962580584075953533656668763

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