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Properties

Label 2-288-96.59-c1-0-11
Degree $2$
Conductor $288$
Sign $0.962 - 0.272i$
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  + (0.290 + 1.38i)2-s + (−1.83 + 0.804i)4-s + (1.12 − 2.71i)5-s + (3.03 − 3.03i)7-s + (−1.64 − 2.29i)8-s + (4.08 + 0.766i)10-s + (−0.616 + 1.48i)11-s + (3.35 − 1.38i)13-s + (5.07 + 3.31i)14-s + (2.70 − 2.94i)16-s − 4.76·17-s + (1.13 + 2.73i)19-s + (0.125 + 5.87i)20-s + (−2.24 − 0.420i)22-s + (−4.11 + 4.11i)23-s + ⋯
L(s)  = 1  + (0.205 + 0.978i)2-s + (−0.915 + 0.402i)4-s + (0.502 − 1.21i)5-s + (1.14 − 1.14i)7-s + (−0.582 − 0.813i)8-s + (1.29 + 0.242i)10-s + (−0.185 + 0.449i)11-s + (0.929 − 0.385i)13-s + (1.35 + 0.885i)14-s + (0.676 − 0.736i)16-s − 1.15·17-s + (0.259 + 0.627i)19-s + (0.0281 + 1.31i)20-s + (−0.477 − 0.0896i)22-s + (−0.857 + 0.857i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.272i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.962 - 0.272i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.962 - 0.272i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49214 + 0.207551i\)
\(L(\frac12)\) \(\approx\) \(1.49214 + 0.207551i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.290 - 1.38i)T \)
3 \( 1 \)
good5 \( 1 + (-1.12 + 2.71i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.03 + 3.03i)T - 7iT^{2} \)
11 \( 1 + (0.616 - 1.48i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (-3.35 + 1.38i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 + 4.76T + 17T^{2} \)
19 \( 1 + (-1.13 - 2.73i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (4.11 - 4.11i)T - 23iT^{2} \)
29 \( 1 + (-8.16 + 3.38i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 - 6.16iT - 31T^{2} \)
37 \( 1 + (9.38 + 3.88i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (-0.169 - 0.169i)T + 41iT^{2} \)
43 \( 1 + (-7.57 - 3.13i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 2.44iT - 47T^{2} \)
53 \( 1 + (1.99 + 0.824i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (1.21 + 0.503i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-1.04 - 2.52i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (-3.91 + 1.62i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (-5.28 - 5.28i)T + 71iT^{2} \)
73 \( 1 + (1.57 - 1.57i)T - 73iT^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + (3.31 - 1.37i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (6.99 - 6.99i)T - 89iT^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−12.14100381598399807207058738239, −10.81074995518032169824710956652, −9.782386115533947451827858926032, −8.626508896870027340707179400387, −8.091958709917571198970727509646, −7.01690589278166249795936987381, −5.72432099252848358290717398213, −4.79685901360836960876453893962, −4.02079105109152421269855129213, −1.29957743769228209657772544575, 1.99581717486370320176670187301, 2.87135742952746657724819718930, 4.44029979344741819342283730504, 5.64863447818565500559664864112, 6.58571887113277499043134698159, 8.370356141003172840607881661567, 8.916041723394953613406685597587, 10.22708140738938202840874028997, 11.02451286525334805386290065099, 11.49103335670691822397292230258

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