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Properties

Label 2-288-72.61-c1-0-1
Degree $2$
Conductor $288$
Sign $-0.376 - 0.926i$
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  + (−0.986 − 1.42i)3-s + (−1.19 + 0.687i)5-s + (−1.80 + 3.12i)7-s + (−1.05 + 2.80i)9-s + (−1.83 − 1.05i)11-s + (−0.887 + 0.512i)13-s + (2.15 + 1.01i)15-s + 0.808·17-s + 7.43i·19-s + (6.23 − 0.512i)21-s + (1.65 + 2.86i)23-s + (−1.55 + 2.69i)25-s + (5.03 − 1.26i)27-s + (−7.71 − 4.45i)29-s + (−3.26 − 5.65i)31-s + ⋯
L(s)  = 1  + (−0.569 − 0.822i)3-s + (−0.532 + 0.307i)5-s + (−0.682 + 1.18i)7-s + (−0.351 + 0.936i)9-s + (−0.552 − 0.319i)11-s + (−0.246 + 0.142i)13-s + (0.556 + 0.262i)15-s + 0.196·17-s + 1.70i·19-s + (1.35 − 0.111i)21-s + (0.345 + 0.597i)23-s + (−0.310 + 0.538i)25-s + (0.969 − 0.243i)27-s + (−1.43 − 0.826i)29-s + (−0.586 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $-0.376 - 0.926i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ -0.376 - 0.926i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.239443 + 0.355680i\)
\(L(\frac12)\) \(\approx\) \(0.239443 + 0.355680i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.986 + 1.42i)T \)
good5 \( 1 + (1.19 - 0.687i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.80 - 3.12i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.83 + 1.05i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.887 - 0.512i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 0.808T + 17T^{2} \)
19 \( 1 - 7.43iT - 19T^{2} \)
23 \( 1 + (-1.65 - 2.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (7.71 + 4.45i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.26 + 5.65i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 4.01iT - 37T^{2} \)
41 \( 1 + (3.45 + 5.99i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.245 - 0.142i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.61 + 6.25i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.86iT - 53T^{2} \)
59 \( 1 + (7.06 - 4.08i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.31 - 3.64i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.43 + 1.40i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.69T + 71T^{2} \)
73 \( 1 - 0.409T + 73T^{2} \)
79 \( 1 + (-0.0456 + 0.0790i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.40 - 1.39i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.91T + 89T^{2} \)
97 \( 1 + (2.76 - 4.78i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.01516463498142065257110220856, −11.49020520197143834931240616730, −10.35936416308677729722900271215, −9.231158180843172307041188234428, −8.020919440527207227851653218172, −7.31958245732542334617116421674, −5.98597931449657369059269263030, −5.49748714484184897900169303508, −3.55978238242241920940214865570, −2.14065165642454899992446469568, 0.32835202483466783529149930194, 3.22237342658950616051327765265, 4.32884038756777234085005948199, 5.18201237080532086525700621812, 6.64973647310137126625779103068, 7.47401497624043690672717943796, 8.874619561767714855573854204696, 9.775867858458445223544674162441, 10.65428975834415919834393501616, 11.25329684459705657005722809157

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