Properties

Degree $2$
Conductor $288$
Sign $0.483 - 0.875i$
Motivic weight $2$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.345 − 1.96i)2-s + (−3.76 + 1.36i)4-s + (−7.20 − 2.98i)5-s + (4.26 − 4.26i)7-s + (3.98 + 6.93i)8-s + (−3.38 + 15.2i)10-s + (−6.19 − 2.56i)11-s + (−8.05 + 3.33i)13-s + (−9.86 − 6.92i)14-s + (12.2 − 10.2i)16-s + 24.5i·17-s + (4.96 + 11.9i)19-s + (31.1 + 1.40i)20-s + (−2.91 + 13.0i)22-s + (9.72 + 9.72i)23-s + ⋯
L(s)  = 1  + (−0.172 − 0.984i)2-s + (−0.940 + 0.340i)4-s + (−1.44 − 0.596i)5-s + (0.608 − 0.608i)7-s + (0.498 + 0.867i)8-s + (−0.338 + 1.52i)10-s + (−0.563 − 0.233i)11-s + (−0.619 + 0.256i)13-s + (−0.704 − 0.494i)14-s + (0.767 − 0.640i)16-s + 1.44i·17-s + (0.261 + 0.630i)19-s + (1.55 + 0.0700i)20-s + (−0.132 + 0.595i)22-s + (0.422 + 0.422i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.483 - 0.875i$
Motivic weight: \(2\)
Character: $\chi_{288} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.483 - 0.875i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.249158 + 0.146942i\)
\(L(\frac12)\) \(\approx\) \(0.249158 + 0.146942i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.345 + 1.96i)T \)
3 \( 1 \)
good5 \( 1 + (7.20 + 2.98i)T + (17.6 + 17.6i)T^{2} \)
7 \( 1 + (-4.26 + 4.26i)T - 49iT^{2} \)
11 \( 1 + (6.19 + 2.56i)T + (85.5 + 85.5i)T^{2} \)
13 \( 1 + (8.05 - 3.33i)T + (119. - 119. i)T^{2} \)
17 \( 1 - 24.5iT - 289T^{2} \)
19 \( 1 + (-4.96 - 11.9i)T + (-255. + 255. i)T^{2} \)
23 \( 1 + (-9.72 - 9.72i)T + 529iT^{2} \)
29 \( 1 + (-5.86 - 14.1i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 17.5iT - 961T^{2} \)
37 \( 1 + (36.0 + 14.9i)T + (968. + 968. i)T^{2} \)
41 \( 1 + (10.9 - 10.9i)T - 1.68e3iT^{2} \)
43 \( 1 + (22.4 + 9.27i)T + (1.30e3 + 1.30e3i)T^{2} \)
47 \( 1 + 27.0T + 2.20e3T^{2} \)
53 \( 1 + (-34.0 + 82.1i)T + (-1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + (27.8 - 67.2i)T + (-2.46e3 - 2.46e3i)T^{2} \)
61 \( 1 + (6.37 + 15.3i)T + (-2.63e3 + 2.63e3i)T^{2} \)
67 \( 1 + (99.2 - 41.0i)T + (3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (-2.55 + 2.55i)T - 5.04e3iT^{2} \)
73 \( 1 + (-30.7 + 30.7i)T - 5.32e3iT^{2} \)
79 \( 1 + 90.6T + 6.24e3T^{2} \)
83 \( 1 + (-39.3 - 94.9i)T + (-4.87e3 + 4.87e3i)T^{2} \)
89 \( 1 + (109. + 109. i)T + 7.92e3iT^{2} \)
97 \( 1 - 63.7T + 9.40e3T^{2} \)
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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

−11.75734255898637641542380098286, −10.89126096892618363351649581905, −10.16788693535688555204211484879, −8.744783249641021219392102829291, −8.128862278391435753872766335417, −7.33452366105132760471954166888, −5.21239271747647522517164276373, −4.28210463061065804886020439310, −3.39993789785810269986106150585, −1.44161688080758487549136583597, 0.16321069218182990644926136018, 2.93620674577707958998320509166, 4.50588858283050224258996841562, 5.26618923223578628903080072232, 6.84428483043061157124043576024, 7.53356013864119426918775540335, 8.238560820825208517284565789496, 9.281314158888209323086918020405, 10.45275138149685952076249172389, 11.51293812185518667735247692836

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