Properties

Degree $2$
Conductor $288$
Sign $0.954 + 0.298i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.658 − 1.88i)2-s + (−3.13 − 2.48i)4-s + (0.659 + 1.59i)5-s + (9.54 + 9.54i)7-s + (−6.75 + 4.27i)8-s + (3.44 − 0.197i)10-s + (3.96 + 9.57i)11-s + (1.91 − 4.63i)13-s + (24.3 − 11.7i)14-s + (3.62 + 15.5i)16-s + 15.3i·17-s + (0.827 + 0.342i)19-s + (1.89 − 6.62i)20-s + (20.6 − 1.18i)22-s + (12.9 − 12.9i)23-s + ⋯
L(s)  = 1  + (0.329 − 0.944i)2-s + (−0.783 − 0.621i)4-s + (0.131 + 0.318i)5-s + (1.36 + 1.36i)7-s + (−0.844 + 0.534i)8-s + (0.344 − 0.0197i)10-s + (0.360 + 0.870i)11-s + (0.147 − 0.356i)13-s + (1.73 − 0.838i)14-s + (0.226 + 0.973i)16-s + 0.900i·17-s + (0.0435 + 0.0180i)19-s + (0.0946 − 0.331i)20-s + (0.940 − 0.0538i)22-s + (0.561 − 0.561i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.954 + 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.954 + 0.298i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.954 + 0.298i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.99014 - 0.303800i\)
\(L(\frac12)\) \(\approx\) \(1.99014 - 0.303800i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.658 + 1.88i)T \)
3 \( 1 \)
good5 \( 1 + (-0.659 - 1.59i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-9.54 - 9.54i)T + 49iT^{2} \)
11 \( 1 + (-3.96 - 9.57i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (-1.91 + 4.63i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 15.3iT - 289T^{2} \)
19 \( 1 + (-0.827 - 0.342i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (-12.9 + 12.9i)T - 529iT^{2} \)
29 \( 1 + (23.7 + 9.85i)T + (594. + 594. i)T^{2} \)
31 \( 1 + 25.1iT - 961T^{2} \)
37 \( 1 + (-13.6 - 32.8i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (-32.9 - 32.9i)T + 1.68e3iT^{2} \)
43 \( 1 + (17.9 + 43.3i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 20.1T + 2.20e3T^{2} \)
53 \( 1 + (35.0 - 14.5i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (-60.6 + 25.1i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (27.9 + 11.5i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (-1.13 + 2.73i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-45.6 - 45.6i)T + 5.04e3iT^{2} \)
73 \( 1 + (29.1 + 29.1i)T + 5.32e3iT^{2} \)
79 \( 1 - 3.27T + 6.24e3T^{2} \)
83 \( 1 + (56.7 + 23.5i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-44.5 + 44.5i)T - 7.92e3iT^{2} \)
97 \( 1 + 106.T + 9.40e3T^{2} \)
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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

−11.54798759691998073643668484185, −10.85307381768810740447854384522, −9.798557880554010328240942419834, −8.810585450119664499165781029968, −8.010568059085675364446920989009, −6.26566654340447210020355681064, −5.24292407851656539060439336733, −4.30634907550360213789252319712, −2.65265861939110598737120251403, −1.66122909696124752080713467339, 1.05388324308823888419799970288, 3.56314509080880119325341926122, 4.64002800711067479334436909194, 5.46596021878380159236340781638, 6.91104711987393207871719919878, 7.58685427563145115484926215075, 8.578454328426971769856355373949, 9.433141708791558502012480182497, 10.91233402284113644022947818996, 11.52979549527659013020425952009

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