Properties

Label 2-768-24.11-c3-0-49
Degree $2$
Conductor $768$
Sign $0.902 + 0.430i$
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  + (−1.73 − 4.89i)3-s + 16.9·5-s − 17.3i·7-s + (−20.9 + 16.9i)9-s + 29.3i·11-s + 26i·13-s + (−29.3 − 83.1i)15-s + 67.8i·17-s + 107.·19-s + (−84.8 + 30i)21-s + 176.·23-s + 162.·25-s + (119. + 73.4i)27-s + 16.9·29-s + 31.1i·31-s + ⋯
L(s)  = 1  + (−0.333 − 0.942i)3-s + 1.51·5-s − 0.935i·7-s + (−0.777 + 0.628i)9-s + 0.805i·11-s + 0.554i·13-s + (−0.505 − 1.43i)15-s + 0.968i·17-s + 1.29·19-s + (−0.881 + 0.311i)21-s + 1.59·23-s + 1.30·25-s + (0.851 + 0.523i)27-s + 0.108·29-s + 0.180i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.902 + 0.430i$
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: \((2,\ 768,\ (\ :3/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.540338413\)
\(L(\frac12)\) \(\approx\) \(2.540338413\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.73 + 4.89i)T \)
good5 \( 1 - 16.9T + 125T^{2} \)
7 \( 1 + 17.3iT - 343T^{2} \)
11 \( 1 - 29.3iT - 1.33e3T^{2} \)
13 \( 1 - 26iT - 2.19e3T^{2} \)
17 \( 1 - 67.8iT - 4.91e3T^{2} \)
19 \( 1 - 107.T + 6.85e3T^{2} \)
23 \( 1 - 176.T + 1.21e4T^{2} \)
29 \( 1 - 16.9T + 2.43e4T^{2} \)
31 \( 1 - 31.1iT - 2.97e4T^{2} \)
37 \( 1 - 206iT - 5.06e4T^{2} \)
41 \( 1 - 305. iT - 6.89e4T^{2} \)
43 \( 1 + 93.5T + 7.95e4T^{2} \)
47 \( 1 - 117.T + 1.03e5T^{2} \)
53 \( 1 + 50.9T + 1.48e5T^{2} \)
59 \( 1 + 558. iT - 2.05e5T^{2} \)
61 \( 1 + 278iT - 2.26e5T^{2} \)
67 \( 1 + 890.T + 3.00e5T^{2} \)
71 \( 1 - 58.7T + 3.57e5T^{2} \)
73 \( 1 - 422T + 3.89e5T^{2} \)
79 \( 1 - 668. iT - 4.93e5T^{2} \)
83 \( 1 - 29.3iT - 5.71e5T^{2} \)
89 \( 1 - 373. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3T + 9.12e5T^{2} \)
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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

−9.899481185335980269377401376412, −9.155641072488126463787801067706, −7.988703665959783277138591112356, −7.00552904381253942876882638652, −6.54504750645395006103915439129, −5.52536878321654782705799184898, −4.68708965591181392388378959268, −3.02712982249250521792361057842, −1.77716979605161011314923304723, −1.11221382676105538071999358852, 0.856156428748142263018517803663, 2.53678145110148456379847488193, 3.25920201254246930586983957242, 4.95105742543954275210764789828, 5.54852235021856806034189323681, 6.01006332362498001819055727454, 7.28184186325353885331596058569, 8.948723406728638731272852307938, 9.023736148189854061188559544570, 9.946974784950696738477434099094

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