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Properties

Label 2-288-36.11-c1-0-8
Degree $2$
Conductor $288$
Sign $0.217 + 0.976i$
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  + (−1.55 − 0.765i)3-s + (1.81 − 1.04i)5-s + (−0.143 − 0.0829i)7-s + (1.82 + 2.37i)9-s + (0.784 − 1.35i)11-s + (−1.93 − 3.34i)13-s + (−3.62 + 0.239i)15-s − 5.27i·17-s − 8.05i·19-s + (0.159 + 0.238i)21-s + (2.67 + 4.63i)23-s + (−0.298 + 0.516i)25-s + (−1.02 − 5.09i)27-s + (6.75 + 3.89i)29-s + (2.10 − 1.21i)31-s + ⋯
L(s)  = 1  + (−0.897 − 0.441i)3-s + (0.812 − 0.469i)5-s + (−0.0543 − 0.0313i)7-s + (0.609 + 0.792i)9-s + (0.236 − 0.409i)11-s + (−0.535 − 0.928i)13-s + (−0.936 + 0.0618i)15-s − 1.27i·17-s − 1.84i·19-s + (0.0348 + 0.0521i)21-s + (0.557 + 0.966i)23-s + (−0.0596 + 0.103i)25-s + (−0.196 − 0.980i)27-s + (1.25 + 0.724i)29-s + (0.378 − 0.218i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.217 + 0.976i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.217 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.819155 - 0.656776i\)
\(L(\frac12)\) \(\approx\) \(0.819155 - 0.656776i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.55 + 0.765i)T \)
good5 \( 1 + (-1.81 + 1.04i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (0.143 + 0.0829i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.784 + 1.35i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.93 + 3.34i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.27iT - 17T^{2} \)
19 \( 1 + 8.05iT - 19T^{2} \)
23 \( 1 + (-2.67 - 4.63i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.75 - 3.89i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.10 + 1.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 8.53T + 37T^{2} \)
41 \( 1 + (-2.47 + 1.43i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.42 - 1.97i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.68 - 6.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.40iT - 53T^{2} \)
59 \( 1 + (-5.49 - 9.52i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.11 - 12.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.45 - 0.841i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.5T + 71T^{2} \)
73 \( 1 - 10.4T + 73T^{2} \)
79 \( 1 + (-6.31 - 3.64i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.35 + 2.34i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 2.40iT - 89T^{2} \)
97 \( 1 + (-0.903 + 1.56i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.62480236892468107892632032484, −10.79474028637943220771055583982, −9.760336072575573822129253294139, −8.900590062178062550262964710336, −7.48033240126462126630787586061, −6.65593590783120857207236419323, −5.41625115928730394716497544718, −4.90300754039069924963449236394, −2.74825274369299320776378949275, −0.940603907024333519210819879898, 1.89741449801155659251601792278, 3.82165934160002506395332315657, 4.96251418851575485920869455744, 6.24180146447006809993197558396, 6.63410683374604825113412803471, 8.225037560075663578835126571539, 9.583783443007420234823408196060, 10.16499883344413644560728879285, 10.87763831268671496409551353265, 12.16391183568653417674434663801

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