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Properties

Label 2-288-9.2-c2-0-11
Degree $2$
Conductor $288$
Sign $0.934 + 0.354i$
Analytic cond. $7.84743$
Root an. cond. $2.80132$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 − 1.04i)3-s + (5.41 − 3.12i)5-s + (−3.74 + 6.49i)7-s + (6.81 + 5.87i)9-s + (6.17 + 3.56i)11-s + (−0.888 − 1.53i)13-s + (−18.4 + 3.13i)15-s − 14.7i·17-s + 19.9·19-s + (17.3 − 14.3i)21-s + (20.8 − 12.0i)23-s + (7.03 − 12.1i)25-s + (−13.0 − 23.6i)27-s + (40.1 + 23.1i)29-s + (−14.1 − 24.5i)31-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)3-s + (1.08 − 0.625i)5-s + (−0.535 + 0.927i)7-s + (0.757 + 0.653i)9-s + (0.561 + 0.324i)11-s + (−0.0683 − 0.118i)13-s + (−1.23 + 0.208i)15-s − 0.869i·17-s + 1.04·19-s + (0.825 − 0.682i)21-s + (0.907 − 0.523i)23-s + (0.281 − 0.487i)25-s + (−0.482 − 0.876i)27-s + (1.38 + 0.798i)29-s + (−0.456 − 0.790i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.934 + 0.354i$
Analytic conductor: \(7.84743\)
Root analytic conductor: \(2.80132\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{288} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 288,\ (\ :1),\ 0.934 + 0.354i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42217 - 0.260771i\)
\(L(\frac12)\) \(\approx\) \(1.42217 - 0.260771i\)
\(L(2)\) 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 \( 1 + (2.81 + 1.04i)T \)
good5 \( 1 + (-5.41 + 3.12i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (3.74 - 6.49i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-6.17 - 3.56i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (0.888 + 1.53i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 14.7iT - 289T^{2} \)
19 \( 1 - 19.9T + 361T^{2} \)
23 \( 1 + (-20.8 + 12.0i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-40.1 - 23.1i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.1 + 24.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 63.0T + 1.36e3T^{2} \)
41 \( 1 + (28.0 - 16.1i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (-38.8 + 67.2i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (-38.4 - 22.2i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 42.2iT - 2.80e3T^{2} \)
59 \( 1 + (93.8 - 54.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (25.3 - 43.9i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (56.9 + 98.6i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 - 85.2iT - 5.04e3T^{2} \)
73 \( 1 + 94.5T + 5.32e3T^{2} \)
79 \( 1 + (-35.6 + 61.6i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-94.9 - 54.7i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 29.6iT - 7.92e3T^{2} \)
97 \( 1 + (62.4 - 108. i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−11.81015586360839682204201447287, −10.58729262002828116767313447089, −9.519810981256733154125260767836, −9.042557871937666550138811711268, −7.42295132015873651512463988762, −6.34925118077870484569151558171, −5.58191229584077163814916352887, −4.74282018677337537303006072992, −2.60032140363171133755219369206, −1.09066620401033225392851615133, 1.15740785650990360482529782694, 3.20937169177216489138290991651, 4.49512686306048795451869921674, 5.88795812030544653259841044477, 6.45664382026958002031491287332, 7.42850810213899421403445655679, 9.214609289445847905095737945255, 9.963201357934221138554021357790, 10.60392301625098258888257806563, 11.42868618688126844277385034631

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