Properties

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

Origins

Related objects

Downloads

Learn more about

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: $\chi_{288} (19, \cdot )$
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 \( 1 - 63.7T + 9.40e3T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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