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Properties

Label 2-288-288.11-c1-0-29
Degree $2$
Conductor $288$
Sign $0.962 - 0.270i$
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  + (1.10 + 0.879i)2-s + (1.40 − 1.01i)3-s + (0.452 + 1.94i)4-s + (−0.419 − 3.18i)5-s + (2.44 + 0.110i)6-s + (0.973 + 3.63i)7-s + (−1.21 + 2.55i)8-s + (0.940 − 2.84i)9-s + (2.33 − 3.89i)10-s + (−1.02 + 0.786i)11-s + (2.61 + 2.27i)12-s + (1.17 − 1.52i)13-s + (−2.11 + 4.88i)14-s + (−3.81 − 4.04i)15-s + (−3.58 + 1.76i)16-s − 2.44·17-s + ⋯
L(s)  = 1  + (0.783 + 0.621i)2-s + (0.810 − 0.585i)3-s + (0.226 + 0.974i)4-s + (−0.187 − 1.42i)5-s + (0.998 + 0.0451i)6-s + (0.368 + 1.37i)7-s + (−0.428 + 0.903i)8-s + (0.313 − 0.949i)9-s + (0.738 − 1.23i)10-s + (−0.308 + 0.237i)11-s + (0.754 + 0.656i)12-s + (0.325 − 0.424i)13-s + (−0.566 + 1.30i)14-s + (−0.986 − 1.04i)15-s + (−0.897 + 0.441i)16-s − 0.592·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.962 - 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.962 - 0.270i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.962 - 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.35101 + 0.324185i\)
\(L(\frac12)\) \(\approx\) \(2.35101 + 0.324185i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.10 - 0.879i)T \)
3 \( 1 + (-1.40 + 1.01i)T \)
good5 \( 1 + (0.419 + 3.18i)T + (-4.82 + 1.29i)T^{2} \)
7 \( 1 + (-0.973 - 3.63i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.02 - 0.786i)T + (2.84 - 10.6i)T^{2} \)
13 \( 1 + (-1.17 + 1.52i)T + (-3.36 - 12.5i)T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 + (-5.23 + 2.16i)T + (13.4 - 13.4i)T^{2} \)
23 \( 1 + (1.73 + 0.465i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (6.70 + 0.883i)T + (28.0 + 7.50i)T^{2} \)
31 \( 1 + (8.25 - 4.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.39 - 10.6i)T + (-26.1 - 26.1i)T^{2} \)
41 \( 1 + (0.643 - 2.40i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.28 + 2.98i)T + (-11.1 + 41.5i)T^{2} \)
47 \( 1 + (-0.888 - 0.512i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.57 + 8.62i)T + (-37.4 - 37.4i)T^{2} \)
59 \( 1 + (-11.3 + 1.49i)T + (56.9 - 15.2i)T^{2} \)
61 \( 1 + (0.590 - 4.48i)T + (-58.9 - 15.7i)T^{2} \)
67 \( 1 + (-0.582 + 0.759i)T + (-17.3 - 64.7i)T^{2} \)
71 \( 1 + (-8.30 - 8.30i)T + 71iT^{2} \)
73 \( 1 + (-11.7 + 11.7i)T - 73iT^{2} \)
79 \( 1 + (-2.72 + 4.71i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.31 + 0.831i)T + (80.1 + 21.4i)T^{2} \)
89 \( 1 + (2.10 - 2.10i)T - 89iT^{2} \)
97 \( 1 + (-2.69 + 4.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.16705226637340283799184920546, −11.55266820335760647753234704937, −9.393724127970519530386343243056, −8.636036159026594499687317530065, −8.156757032043021600335439933897, −7.02876824386533062815093453521, −5.60872494577157838974309186527, −4.94456136673621517934537193896, −3.45590544004685724393436047853, −2.01059925563928709534972185258, 2.13054810611463761664775424572, 3.57907383480530099651576641789, 3.93754378020329736606221202123, 5.48884948967598268936162637163, 7.02775188826845538599937146271, 7.61234835825677356822583604769, 9.300278663014548470876839546106, 10.22174797500066476811365943603, 10.94725965464834286436611716452, 11.29604577971595064007415648823

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