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Properties

Label 2-768-24.11-c3-0-2
Degree $2$
Conductor $768$
Sign $-0.987 - 0.159i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.21 − 3.04i)3-s − 9.33·5-s + 36.3i·7-s + (8.50 − 25.6i)9-s + 48.4i·11-s − 25.8i·13-s + (−39.3 + 28.3i)15-s − 74.2i·17-s + 82.9·19-s + (110. + 153. i)21-s − 179.·23-s − 37.8·25-s + (−42.1 − 133. i)27-s − 122.·29-s + 64.1i·31-s + ⋯
L(s)  = 1  + (0.810 − 0.585i)3-s − 0.834·5-s + 1.96i·7-s + (0.314 − 0.949i)9-s + 1.32i·11-s − 0.552i·13-s + (−0.676 + 0.488i)15-s − 1.05i·17-s + 1.00·19-s + (1.14 + 1.59i)21-s − 1.62·23-s − 0.303·25-s + (−0.300 − 0.953i)27-s − 0.783·29-s + 0.371i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.987 - 0.159i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ -0.987 - 0.159i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.3246780742\)
\(L(\frac12)\) \(\approx\) \(0.3246780742\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.21 + 3.04i)T \)
good5 \( 1 + 9.33T + 125T^{2} \)
7 \( 1 - 36.3iT - 343T^{2} \)
11 \( 1 - 48.4iT - 1.33e3T^{2} \)
13 \( 1 + 25.8iT - 2.19e3T^{2} \)
17 \( 1 + 74.2iT - 4.91e3T^{2} \)
19 \( 1 - 82.9T + 6.85e3T^{2} \)
23 \( 1 + 179.T + 1.21e4T^{2} \)
29 \( 1 + 122.T + 2.43e4T^{2} \)
31 \( 1 - 64.1iT - 2.97e4T^{2} \)
37 \( 1 - 5.01iT - 5.06e4T^{2} \)
41 \( 1 + 325. iT - 6.89e4T^{2} \)
43 \( 1 + 321.T + 7.95e4T^{2} \)
47 \( 1 + 95.9T + 1.03e5T^{2} \)
53 \( 1 - 185.T + 1.48e5T^{2} \)
59 \( 1 - 226. iT - 2.05e5T^{2} \)
61 \( 1 + 198. iT - 2.26e5T^{2} \)
67 \( 1 + 23.9T + 3.00e5T^{2} \)
71 \( 1 + 399.T + 3.57e5T^{2} \)
73 \( 1 + 669.T + 3.89e5T^{2} \)
79 \( 1 - 229. iT - 4.93e5T^{2} \)
83 \( 1 - 321. iT - 5.71e5T^{2} \)
89 \( 1 + 131. iT - 7.04e5T^{2} \)
97 \( 1 - 136.T + 9.12e5T^{2} \)
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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

−9.966923877204743571228398056317, −9.372977083949229860854610380279, −8.557757250046638534069552282273, −7.78434531783814475056195465229, −7.15782138072906672285286684559, −5.93761929203831036500828923005, −5.00349748011227719832669355167, −3.67698157652848125419849524352, −2.64882518252887458581467759238, −1.83382173446204280374501024847, 0.07508105710161636313119954100, 1.51074460813933017992356776137, 3.37561018996525253189410955435, 3.81829993042165840265039403653, 4.52698666160180634615451009709, 6.01960847075378820738445483154, 7.25022539241886940222469396694, 7.908671611205851301077802781407, 8.435700091567816030519358132264, 9.701454224824636261285376023307

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