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Properties

Label 2-288-72.59-c1-0-6
Degree $2$
Conductor $288$
Sign $0.950 - 0.310i$
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  + (1.12 + 1.31i)3-s + (0.565 − 0.978i)5-s + (3.71 − 2.14i)7-s + (−0.456 + 2.96i)9-s + (−1.00 + 0.582i)11-s + (−2.64 − 1.52i)13-s + (1.92 − 0.360i)15-s − 1.49i·17-s + 3.42·19-s + (7.00 + 2.46i)21-s + (−3.85 + 6.68i)23-s + (1.86 + 3.22i)25-s + (−4.41 + 2.74i)27-s + (0.709 + 1.22i)29-s + (−4.66 − 2.69i)31-s + ⋯
L(s)  = 1  + (0.651 + 0.758i)3-s + (0.252 − 0.437i)5-s + (1.40 − 0.810i)7-s + (−0.152 + 0.988i)9-s + (−0.304 + 0.175i)11-s + (−0.733 − 0.423i)13-s + (0.496 − 0.0932i)15-s − 0.362i·17-s + 0.785·19-s + (1.52 + 0.537i)21-s + (−0.804 + 1.39i)23-s + (0.372 + 0.644i)25-s + (−0.849 + 0.528i)27-s + (0.131 + 0.228i)29-s + (−0.837 − 0.483i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.310i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.950 - 0.310i$
Analytic conductor: \(2.29969\)
Root analytic conductor: \(1.51647\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1/2),\ 0.950 - 0.310i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72407 + 0.274302i\)
\(L(\frac12)\) \(\approx\) \(1.72407 + 0.274302i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.12 - 1.31i)T \)
good5 \( 1 + (-0.565 + 0.978i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-3.71 + 2.14i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.00 - 0.582i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.64 + 1.52i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + 1.49iT - 17T^{2} \)
19 \( 1 - 3.42T + 19T^{2} \)
23 \( 1 + (3.85 - 6.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.709 - 1.22i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.66 + 2.69i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.97iT - 37T^{2} \)
41 \( 1 + (4.23 + 2.44i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.74 - 3.01i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.77 + 3.08i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 + (7.50 + 4.33i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.16 + 1.82i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.58 + 9.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.54T + 71T^{2} \)
73 \( 1 + 7.06T + 73T^{2} \)
79 \( 1 + (2.24 - 1.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.98 + 2.30i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 8.63iT - 89T^{2} \)
97 \( 1 + (-3.35 - 5.81i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.62561685013285258149611684268, −10.84615152969253644155677289373, −9.896527458508421140118735983565, −9.139390600521310914295417644993, −7.86480293502678013526351415360, −7.51578997250138475698121608679, −5.33267142219272103542616041464, −4.79702717389723364743210261208, −3.50338275155831763215564640564, −1.80925375524667014239409419624, 1.82455123387258135573811726061, 2.81331566885800559844710971885, 4.61059836408489319090501536149, 5.87372769722063187798468599264, 6.99400485664105307681294993842, 8.049460094662255710184210077973, 8.620525443546448246730183068679, 9.777787887542737815744161008710, 10.95499876476115814827863056578, 11.95217196717679549621067332654

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