Properties

Degree $2$
Conductor $288$
Sign $-0.933 - 0.358i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.682 + 1.87i)2-s + (−3.06 − 2.56i)4-s + (−1.34 − 3.25i)5-s + (0.583 + 0.583i)7-s + (6.91 − 4.01i)8-s + (7.03 − 0.313i)10-s + (3.03 + 7.33i)11-s + (−6.38 + 15.4i)13-s + (−1.49 + 0.698i)14-s + (2.83 + 15.7i)16-s + 19.0i·17-s + (−29.6 − 12.2i)19-s + (−4.21 + 13.4i)20-s + (−15.8 + 0.706i)22-s + (−15.2 + 15.2i)23-s + ⋯
L(s)  = 1  + (−0.341 + 0.939i)2-s + (−0.767 − 0.641i)4-s + (−0.269 − 0.650i)5-s + (0.0833 + 0.0833i)7-s + (0.864 − 0.502i)8-s + (0.703 − 0.0313i)10-s + (0.276 + 0.666i)11-s + (−0.491 + 1.18i)13-s + (−0.106 + 0.0498i)14-s + (0.177 + 0.984i)16-s + 1.12i·17-s + (−1.56 − 0.646i)19-s + (−0.210 + 0.671i)20-s + (−0.720 + 0.0320i)22-s + (−0.665 + 0.665i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.933 - 0.358i$
Motivic weight: \(2\)
Character: $\chi_{288} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ -0.933 - 0.358i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.125654 + 0.678046i\)
\(L(\frac12)\) \(\approx\) \(0.125654 + 0.678046i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.682 - 1.87i)T \)
3 \( 1 \)
good5 \( 1 + (1.34 + 3.25i)T + (-17.6 + 17.6i)T^{2} \)
7 \( 1 + (-0.583 - 0.583i)T + 49iT^{2} \)
11 \( 1 + (-3.03 - 7.33i)T + (-85.5 + 85.5i)T^{2} \)
13 \( 1 + (6.38 - 15.4i)T + (-119. - 119. i)T^{2} \)
17 \( 1 - 19.0iT - 289T^{2} \)
19 \( 1 + (29.6 + 12.2i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (15.2 - 15.2i)T - 529iT^{2} \)
29 \( 1 + (-20.5 - 8.49i)T + (594. + 594. i)T^{2} \)
31 \( 1 - 53.6iT - 961T^{2} \)
37 \( 1 + (3.80 + 9.17i)T + (-968. + 968. i)T^{2} \)
41 \( 1 + (14.5 + 14.5i)T + 1.68e3iT^{2} \)
43 \( 1 + (-20.3 - 49.1i)T + (-1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 4.73T + 2.20e3T^{2} \)
53 \( 1 + (61.4 - 25.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (42.4 - 17.5i)T + (2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (27.7 + 11.4i)T + (2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (9.42 - 22.7i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-95.1 - 95.1i)T + 5.04e3iT^{2} \)
73 \( 1 + (-37.1 - 37.1i)T + 5.32e3iT^{2} \)
79 \( 1 + 70.3T + 6.24e3T^{2} \)
83 \( 1 + (14.5 + 6.01i)T + (4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (-60.8 + 60.8i)T - 7.92e3iT^{2} \)
97 \( 1 - 31.8T + 9.40e3T^{2} \)
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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

−12.24500745660851952661656521277, −10.87601231083972493588044091663, −9.876715104777154206398237892393, −8.874794655362293548577971087002, −8.320851300171795875246662092591, −7.05687518149364421695210789553, −6.30463904971796747533867711411, −4.85706550598082001180402501140, −4.19796563097136879049340079648, −1.67808264758379149817156949198, 0.38559615796783773432894069767, 2.40845239620308339251575520052, 3.46925013900845171054294531580, 4.70396680110928208547056006763, 6.20335014195142239505533501889, 7.60978847239797618209731405269, 8.322994904521627281548993455968, 9.486950510066394893997626796151, 10.45129642064382998988297827158, 11.01843831568901085303744844902

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