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Properties

Label 2-288-96.59-c1-0-5
Degree $2$
Conductor $288$
Sign $-0.781 - 0.624i$
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  + (0.850 + 1.12i)2-s + (−0.553 + 1.92i)4-s + (−0.352 + 0.852i)5-s + (−3.43 + 3.43i)7-s + (−2.64 + 1.00i)8-s + (−1.26 + 0.325i)10-s + (1.44 − 3.49i)11-s + (−0.258 + 0.107i)13-s + (−6.80 − 0.959i)14-s + (−3.38 − 2.12i)16-s + 5.30·17-s + (2.72 + 6.57i)19-s + (−1.44 − 1.14i)20-s + (5.17 − 1.33i)22-s + (−2.23 + 2.23i)23-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)2-s + (−0.276 + 0.960i)4-s + (−0.157 + 0.381i)5-s + (−1.29 + 1.29i)7-s + (−0.934 + 0.356i)8-s + (−0.399 + 0.103i)10-s + (0.436 − 1.05i)11-s + (−0.0717 + 0.0297i)13-s + (−1.81 − 0.256i)14-s + (−0.846 − 0.531i)16-s + 1.28·17-s + (0.625 + 1.50i)19-s + (−0.322 − 0.257i)20-s + (1.10 − 0.284i)22-s + (−0.466 + 0.466i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.781 - 0.624i$
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: \((2,\ 288,\ (\ :1/2),\ -0.781 - 0.624i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.450816 + 1.28571i\)
\(L(\frac12)\) \(\approx\) \(0.450816 + 1.28571i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.850 - 1.12i)T \)
3 \( 1 \)
good5 \( 1 + (0.352 - 0.852i)T + (-3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.43 - 3.43i)T - 7iT^{2} \)
11 \( 1 + (-1.44 + 3.49i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + (0.258 - 0.107i)T + (9.19 - 9.19i)T^{2} \)
17 \( 1 - 5.30T + 17T^{2} \)
19 \( 1 + (-2.72 - 6.57i)T + (-13.4 + 13.4i)T^{2} \)
23 \( 1 + (2.23 - 2.23i)T - 23iT^{2} \)
29 \( 1 + (-3.16 + 1.31i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + 3.46iT - 31T^{2} \)
37 \( 1 + (-1.27 - 0.528i)T + (26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.28 + 5.28i)T + 41iT^{2} \)
43 \( 1 + (-2.46 - 1.02i)T + (30.4 + 30.4i)T^{2} \)
47 \( 1 - 0.423iT - 47T^{2} \)
53 \( 1 + (-12.5 - 5.20i)T + (37.4 + 37.4i)T^{2} \)
59 \( 1 + (-5.24 - 2.17i)T + (41.7 + 41.7i)T^{2} \)
61 \( 1 + (-0.0138 - 0.0333i)T + (-43.1 + 43.1i)T^{2} \)
67 \( 1 + (9.82 - 4.06i)T + (47.3 - 47.3i)T^{2} \)
71 \( 1 + (4.64 + 4.64i)T + 71iT^{2} \)
73 \( 1 + (-3.96 + 3.96i)T - 73iT^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + (-0.867 + 0.359i)T + (58.6 - 58.6i)T^{2} \)
89 \( 1 + (4.82 - 4.82i)T - 89iT^{2} \)
97 \( 1 - 8.78T + 97T^{2} \)
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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

−12.10186591924043321168901040773, −11.82027049709553221844433776177, −10.11326385971473543400971386907, −9.158193195509438889053952775243, −8.261017187864673706680689254001, −7.15013694226854738150287924815, −5.88695235748147196889524854040, −5.71384613268407264645440036901, −3.67648775471702670387474499653, −2.98111687493739808678420885491, 0.892289280163225790929236047248, 2.94727486220125104081816917054, 4.03924196938838770868073692144, 4.98186296306042672591209125137, 6.49802196860806271385223737934, 7.25363662395346036831391287908, 8.943073220683390227354835868380, 10.00481205266915577436749280027, 10.26509804277833329346905587643, 11.64500479184347414825937216736

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