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Properties

Label 2-288-32.5-c1-0-7
Degree $2$
Conductor $288$
Sign $0.812 - 0.583i$
Analytic cond. $2.29969$
Root an. cond. $1.51647$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.06 + 0.925i)2-s + (0.286 − 1.97i)4-s + (3.41 + 1.41i)5-s + (−1.42 − 1.42i)7-s + (1.52 + 2.38i)8-s + (−4.95 + 1.64i)10-s + (0.821 − 1.98i)11-s + (3.77 − 1.56i)13-s + (2.85 + 0.205i)14-s + (−3.83 − 1.13i)16-s + 0.438i·17-s + (2.08 − 0.865i)19-s + (3.77 − 6.35i)20-s + (0.957 + 2.88i)22-s + (−4.42 + 4.42i)23-s + ⋯
L(s)  = 1  + (−0.756 + 0.654i)2-s + (0.143 − 0.989i)4-s + (1.52 + 0.632i)5-s + (−0.540 − 0.540i)7-s + (0.539 + 0.841i)8-s + (−1.56 + 0.521i)10-s + (0.247 − 0.598i)11-s + (1.04 − 0.433i)13-s + (0.762 + 0.0548i)14-s + (−0.958 − 0.283i)16-s + 0.106i·17-s + (0.479 − 0.198i)19-s + (0.844 − 1.42i)20-s + (0.204 + 0.614i)22-s + (−0.922 + 0.922i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.812 - 0.583i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07812 + 0.346851i\)
\(L(\frac12)\) \(\approx\) \(1.07812 + 0.346851i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.06 - 0.925i)T \)
3 \( 1 \)
good5 \( 1 + (-3.41 - 1.41i)T + (3.53 + 3.53i)T^{2} \)
7 \( 1 + (1.42 + 1.42i)T + 7iT^{2} \)
11 \( 1 + (-0.821 + 1.98i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-3.77 + 1.56i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 0.438iT - 17T^{2} \)
19 \( 1 + (-2.08 + 0.865i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (4.42 - 4.42i)T - 23iT^{2} \)
29 \( 1 + (-2.48 - 6.00i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 - 10.5T + 31T^{2} \)
37 \( 1 + (8.11 + 3.36i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (3.71 - 3.71i)T - 41iT^{2} \)
43 \( 1 + (4.47 - 10.8i)T + (-30.4 - 30.4i)T^{2} \)
47 \( 1 + 7.94iT - 47T^{2} \)
53 \( 1 + (-1.97 + 4.77i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (2.54 + 1.05i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (2.65 + 6.42i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (0.876 + 2.11i)T + (-47.3 + 47.3i)T^{2} \)
71 \( 1 + (8.63 + 8.63i)T + 71iT^{2} \)
73 \( 1 + (3.27 - 3.27i)T - 73iT^{2} \)
79 \( 1 + 2.97iT - 79T^{2} \)
83 \( 1 + (6.25 - 2.58i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (-0.0975 - 0.0975i)T + 89iT^{2} \)
97 \( 1 - 7.26T + 97T^{2} \)
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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

−11.56496776092998818963332344509, −10.45044556374974053716310958087, −10.08604150835497965979391574181, −9.138567451797691047935686173991, −8.127870730488257314389456061290, −6.74349965340707455298163638454, −6.26037278739149299843590945306, −5.30398767553707787027695937723, −3.21085832620679054114922501636, −1.46329433623166300908462896824, 1.49539849371755266915658739105, 2.66178543852959398667090539213, 4.35924321802708382419719579265, 5.89061667383755650672115073532, 6.72073591296025978890031611243, 8.377578890146609167358725637087, 9.011694084256880691107815205232, 9.899127426196442477804286996757, 10.35756535383263704563170751635, 11.90885941623922871259069318624

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