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Properties

Label 2-288-9.4-c1-0-0
Degree $2$
Conductor $288$
Sign $0.118 - 0.993i$
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  + (−1.35 − 1.07i)3-s + (−1.18 + 2.05i)5-s + (−1.10 − 1.91i)7-s + (0.686 + 2.92i)9-s + (2.96 + 5.14i)11-s + (−2.18 + 3.78i)13-s + (3.82 − 1.51i)15-s + 3.37·17-s − 3.72·19-s + (−0.558 + 3.78i)21-s + (−1.10 + 1.91i)23-s + (−0.313 − 0.543i)25-s + (2.20 − 4.70i)27-s + (−0.186 − 0.322i)29-s + (−4.83 + 8.36i)31-s + ⋯
L(s)  = 1  + (−0.783 − 0.621i)3-s + (−0.530 + 0.918i)5-s + (−0.417 − 0.723i)7-s + (0.228 + 0.973i)9-s + (0.894 + 1.54i)11-s + (−0.606 + 1.05i)13-s + (0.986 − 0.390i)15-s + 0.817·17-s − 0.854·19-s + (−0.121 + 0.826i)21-s + (−0.230 + 0.399i)23-s + (−0.0627 − 0.108i)25-s + (0.425 − 0.905i)27-s + (−0.0345 − 0.0598i)29-s + (−0.867 + 1.50i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.118 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.118 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.118 - 0.993i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.118 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506497 + 0.449858i\)
\(L(\frac12)\) \(\approx\) \(0.506497 + 0.449858i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.35 + 1.07i)T \)
good5 \( 1 + (1.18 - 2.05i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.10 + 1.91i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2.96 - 5.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.18 - 3.78i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.37T + 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
23 \( 1 + (1.10 - 1.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.186 + 0.322i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.83 - 8.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (0.5 - 0.866i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.96 + 5.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.10 + 1.91i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4T + 53T^{2} \)
59 \( 1 + (5.17 - 8.96i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.55 + 13.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.17 + 8.96i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 4.41T + 71T^{2} \)
73 \( 1 - 4.62T + 73T^{2} \)
79 \( 1 + (-4.83 - 8.36i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.04 - 12.1i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 1.25T + 89T^{2} \)
97 \( 1 + (4.5 + 7.79i)T + (-48.5 + 84.0i)T^{2} \)
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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

−12.11644131765556137760856694616, −11.15385623426028252039796753663, −10.32554181456378380641840553756, −9.430733476862950755500270968022, −7.70443974276699942055059360546, −6.96067524058741009683967352492, −6.57297541432577537221552977626, −4.86437345975321369998086933843, −3.74494744379486701084903876311, −1.85660372514842916522526418044, 0.56742523273415704132844342604, 3.27044708014970140269218717457, 4.42525794540025393337686531930, 5.62873432579705496521888380399, 6.21753520166673551868202955216, 7.942240185508633765502637252104, 8.854757491161289420679433767751, 9.642944795413014060130757751012, 10.78822411605275083068340576924, 11.65886600150296492444399496090

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