Properties

Label 2-288-9.7-c1-0-6
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: $\chi_{288} (97, \cdot )$
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

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

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