Properties

Label 2-288-36.23-c1-0-3
Degree $2$
Conductor $288$
Sign $0.317 - 0.948i$
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  + (0.683 + 1.59i)3-s + (3.40 + 1.96i)5-s + (−0.961 + 0.555i)7-s + (−2.06 + 2.17i)9-s + (−1.63 − 2.82i)11-s + (0.124 − 0.216i)13-s + (−0.802 + 6.77i)15-s − 5.86i·17-s − 2.19i·19-s + (−1.54 − 1.15i)21-s + (−2.79 + 4.83i)23-s + (5.24 + 9.09i)25-s + (−4.87 − 1.80i)27-s + (2.35 − 1.36i)29-s + (8.96 + 5.17i)31-s + ⋯
L(s)  = 1  + (0.394 + 0.918i)3-s + (1.52 + 0.880i)5-s + (−0.363 + 0.209i)7-s + (−0.688 + 0.725i)9-s + (−0.492 − 0.852i)11-s + (0.0346 − 0.0600i)13-s + (−0.207 + 1.74i)15-s − 1.42i·17-s − 0.504i·19-s + (−0.336 − 0.251i)21-s + (−0.582 + 1.00i)23-s + (1.04 + 1.81i)25-s + (−0.938 − 0.346i)27-s + (0.437 − 0.252i)29-s + (1.61 + 0.930i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.317 - 0.948i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.35675 + 0.976413i\)
\(L(\frac12)\) \(\approx\) \(1.35675 + 0.976413i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.683 - 1.59i)T \)
good5 \( 1 + (-3.40 - 1.96i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (0.961 - 0.555i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.63 + 2.82i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.124 + 0.216i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.86iT - 17T^{2} \)
19 \( 1 + 2.19iT - 19T^{2} \)
23 \( 1 + (2.79 - 4.83i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.35 + 1.36i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.96 - 5.17i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.333T + 37T^{2} \)
41 \( 1 + (5.28 + 3.05i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-8.50 + 4.91i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.70 + 8.15i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.75iT - 53T^{2} \)
59 \( 1 + (-3.26 + 5.65i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.07 - 1.85i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.501 + 0.289i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 3.26T + 71T^{2} \)
73 \( 1 + 12.6T + 73T^{2} \)
79 \( 1 + (7.67 - 4.42i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.34 - 4.05i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 4.75iT - 89T^{2} \)
97 \( 1 + (-0.916 - 1.58i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.73408937548965384471928080685, −10.77534426342100168429244244307, −10.00879891636980867043288981042, −9.452205060888429184284574600720, −8.441991876793078023786877369425, −6.97412666291651860134303648170, −5.86340130174568095796888321168, −5.07514108985491078822102697016, −3.23813079028156919898505294644, −2.48601803559889169227182846657, 1.45490871452556509450734378726, 2.54243642907929306029117618217, 4.48758272895218606902435844655, 5.92806138560780667115166707428, 6.45570778207638371577808412303, 7.909609802049652699267231330081, 8.694448420619448056568190452510, 9.762941179429838120174980782687, 10.34810975886908212908332353789, 12.04712435157556089338639045616

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