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Properties

Label 2-288-9.4-c1-0-5
Degree $2$
Conductor $288$
Sign $0.635 - 0.771i$
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 + 1.41i)3-s + (0.5 − 0.866i)5-s + (0.724 + 1.25i)7-s + (−1.00 + 2.82i)9-s + (1.72 + 2.98i)11-s + (1.94 − 3.37i)13-s + (1.72 − 0.158i)15-s − 4.89·17-s + 4·19-s + (−1.05 + 2.28i)21-s + (0.275 − 0.476i)23-s + (2 + 3.46i)25-s + (−5.00 + 1.41i)27-s + (−4.94 − 8.57i)29-s + (−3.72 + 6.45i)31-s + ⋯
L(s)  = 1  + (0.577 + 0.816i)3-s + (0.223 − 0.387i)5-s + (0.273 + 0.474i)7-s + (−0.333 + 0.942i)9-s + (0.520 + 0.900i)11-s + (0.540 − 0.936i)13-s + (0.445 − 0.0410i)15-s − 1.18·17-s + 0.917·19-s + (−0.229 + 0.497i)21-s + (0.0573 − 0.0994i)23-s + (0.400 + 0.692i)25-s + (−0.962 + 0.272i)27-s + (−0.919 − 1.59i)29-s + (−0.668 + 1.15i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.635 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(288\)    =    \(2^{5} \cdot 3^{2}\)
Sign: $0.635 - 0.771i$
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: \((2,\ 288,\ (\ :1/2),\ 0.635 - 0.771i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.46730 + 0.692474i\)
\(L(\frac12)\) \(\approx\) \(1.46730 + 0.692474i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.41i)T \)
good5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.724 - 1.25i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.94 + 3.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-0.275 + 0.476i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.94 + 8.57i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.72 - 6.45i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.89T + 37T^{2} \)
41 \( 1 + (-1.05 + 1.81i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.17 + 10.6i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.17 + 7.22i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 0.898T + 53T^{2} \)
59 \( 1 + (-0.174 + 0.301i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.949 + 1.64i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.17 + 2.03i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 4.89T + 73T^{2} \)
79 \( 1 + (4.27 + 7.40i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.72 - 4.71i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 3.10T + 89T^{2} \)
97 \( 1 + (2.94 + 5.10i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.82782605066980285517700872194, −10.92885317127161543234462025008, −9.889367846441103405604076553281, −9.127414246479911924559155321964, −8.384599591567832178479766498476, −7.22753108179000511519004744035, −5.66970187538746219855186958130, −4.77378150722055495741520014556, −3.57442300361465014986418461469, −2.07552688324018637090115211045, 1.43111160062968314284468391098, 2.97200087401570772926206173816, 4.22314448988996876319154296437, 6.03760891892484447331283984319, 6.78469894815488152792174459391, 7.76080480194947032409279009545, 8.829647466751290307177146738037, 9.516153872317038295425137179922, 11.14877795275595989431313305343, 11.40942559765661940517177777433

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