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Properties

Label 2-768-24.11-c3-0-23
Degree $2$
Conductor $768$
Sign $-0.489 - 0.872i$
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  + (−1.40 + 5.00i)3-s + 5.86·5-s − 5.92i·7-s + (−23.0 − 14.0i)9-s − 27.9i·11-s + 0.0653i·13-s + (−8.26 + 29.3i)15-s + 36.9i·17-s + 30.7·19-s + (29.6 + 8.33i)21-s + 61.2·23-s − 90.5·25-s + (102. − 95.3i)27-s − 143.·29-s + 299. i·31-s + ⋯
L(s)  = 1  + (−0.270 + 0.962i)3-s + 0.524·5-s − 0.319i·7-s + (−0.853 − 0.521i)9-s − 0.766i·11-s + 0.00139i·13-s + (−0.142 + 0.505i)15-s + 0.527i·17-s + 0.371·19-s + (0.307 + 0.0866i)21-s + 0.555·23-s − 0.724·25-s + (0.733 − 0.679i)27-s − 0.919·29-s + 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.489 - 0.872i$
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: \((2,\ 768,\ (\ :3/2),\ -0.489 - 0.872i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.440454854\)
\(L(\frac12)\) \(\approx\) \(1.440454854\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.40 - 5.00i)T \)
good5 \( 1 - 5.86T + 125T^{2} \)
7 \( 1 + 5.92iT - 343T^{2} \)
11 \( 1 + 27.9iT - 1.33e3T^{2} \)
13 \( 1 - 0.0653iT - 2.19e3T^{2} \)
17 \( 1 - 36.9iT - 4.91e3T^{2} \)
19 \( 1 - 30.7T + 6.85e3T^{2} \)
23 \( 1 - 61.2T + 1.21e4T^{2} \)
29 \( 1 + 143.T + 2.43e4T^{2} \)
31 \( 1 - 299. iT - 2.97e4T^{2} \)
37 \( 1 - 340. iT - 5.06e4T^{2} \)
41 \( 1 - 379. iT - 6.89e4T^{2} \)
43 \( 1 - 470.T + 7.95e4T^{2} \)
47 \( 1 - 428.T + 1.03e5T^{2} \)
53 \( 1 + 505.T + 1.48e5T^{2} \)
59 \( 1 - 207. iT - 2.05e5T^{2} \)
61 \( 1 + 578. iT - 2.26e5T^{2} \)
67 \( 1 + 415.T + 3.00e5T^{2} \)
71 \( 1 + 547.T + 3.57e5T^{2} \)
73 \( 1 + 194.T + 3.89e5T^{2} \)
79 \( 1 - 308. iT - 4.93e5T^{2} \)
83 \( 1 - 62.3iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 - 703.T + 9.12e5T^{2} \)
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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

−10.28165757961831435208768902477, −9.408926491599190952274410289328, −8.758420545952210977025830824657, −7.72427193057643367927327663288, −6.45343727407117533863252790640, −5.73233249847398568156925799218, −4.85203996806164782327361247896, −3.78676599481303837415267295569, −2.86302080568435784943442796656, −1.16743186637614268780271655993, 0.43229609011323894403651744478, 1.85039516088338196760032437120, 2.60142226058834307656099961869, 4.19697465830053518112440340584, 5.56919624674706778619977508834, 5.91720370433750340317866902381, 7.34243745370283453233873685094, 7.47885583426877410334000671490, 8.925735910586245636130066127974, 9.471222599082344612911898939999

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