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Properties

Label 2-24e2-9.4-c1-0-15
Degree $2$
Conductor $576$
Sign $0.800 + 0.598i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.800 + 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.800 + 0.598i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.800 + 0.598i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.83809 - 0.611074i\)
\(L(\frac12)\) \(\approx\) \(1.83809 - 0.611074i\)
\(L(\frac{3}{2})\) 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

−10.40299041992337589199393869367, −9.835868518200201621301743606809, −8.763189722158044997742512185317, −8.269525512225082437688857239890, −7.30626221555478622889933213435, −5.60545489758768981104403786348, −5.26171018368509314413668171981, −3.66587732962684528664707314494, −3.04673187316807711314603711760, −1.09706855937034209199280950337, 1.96936100365335829753372842641, 2.65499361137664750720703693196, 3.89201278365343773625378035178, 5.47453928292029581967017420023, 6.50083413254368028495418332395, 7.20586985540956566903236047649, 8.022670890518897278842344224314, 9.279864940255529883970728749071, 9.699703960155807702417459626924, 10.64829828894156356417874311158

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