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Properties

Label 2-768-24.11-c3-0-88
Degree $2$
Conductor $768$
Sign $-0.961 - 0.275i$
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  + (4.54 − 2.52i)3-s − 8.01·5-s − 12.6i·7-s + (14.2 − 22.9i)9-s − 37.7i·11-s + 60.0i·13-s + (−36.3 + 20.2i)15-s − 37.0i·17-s − 127.·19-s + (−31.7 − 57.2i)21-s − 56.9·23-s − 60.8·25-s + (7.09 − 140. i)27-s + 220.·29-s − 2.26i·31-s + ⋯
L(s)  = 1  + (0.874 − 0.485i)3-s − 0.716·5-s − 0.680i·7-s + (0.528 − 0.848i)9-s − 1.03i·11-s + 1.28i·13-s + (−0.626 + 0.347i)15-s − 0.528i·17-s − 1.54·19-s + (−0.330 − 0.594i)21-s − 0.516·23-s − 0.486·25-s + (0.0505 − 0.998i)27-s + 1.41·29-s − 0.0130i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.961 - 0.275i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.961 - 0.275i$
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: \((2,\ 768,\ (\ :3/2),\ -0.961 - 0.275i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6804999647\)
\(L(\frac12)\) \(\approx\) \(0.6804999647\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.54 + 2.52i)T \)
good5 \( 1 + 8.01T + 125T^{2} \)
7 \( 1 + 12.6iT - 343T^{2} \)
11 \( 1 + 37.7iT - 1.33e3T^{2} \)
13 \( 1 - 60.0iT - 2.19e3T^{2} \)
17 \( 1 + 37.0iT - 4.91e3T^{2} \)
19 \( 1 + 127.T + 6.85e3T^{2} \)
23 \( 1 + 56.9T + 1.21e4T^{2} \)
29 \( 1 - 220.T + 2.43e4T^{2} \)
31 \( 1 + 2.26iT - 2.97e4T^{2} \)
37 \( 1 - 166. iT - 5.06e4T^{2} \)
41 \( 1 - 154. iT - 6.89e4T^{2} \)
43 \( 1 - 53.4T + 7.95e4T^{2} \)
47 \( 1 + 591.T + 1.03e5T^{2} \)
53 \( 1 + 538.T + 1.48e5T^{2} \)
59 \( 1 + 586. iT - 2.05e5T^{2} \)
61 \( 1 - 431. iT - 2.26e5T^{2} \)
67 \( 1 - 175.T + 3.00e5T^{2} \)
71 \( 1 - 29.5T + 3.57e5T^{2} \)
73 \( 1 + 937.T + 3.89e5T^{2} \)
79 \( 1 + 409. iT - 4.93e5T^{2} \)
83 \( 1 - 1.29e3iT - 5.71e5T^{2} \)
89 \( 1 + 921. iT - 7.04e5T^{2} \)
97 \( 1 + 1.64e3T + 9.12e5T^{2} \)
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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

−9.310998912203974806689562880264, −8.375029227797196139889417136726, −7.977977293165554019731173012559, −6.82831255606303589385757405642, −6.34551593252668040286886723712, −4.53440727648142391140705660405, −3.87649195327924893184634586104, −2.82341081086041772178661773703, −1.50557506485881272274930772177, −0.15467224628908288893633255758, 1.90992007401793941483229833407, 2.89861558646154853337291228139, 3.99927181789660511239611859074, 4.74218116552259427632798772701, 5.93552511071652839218625793118, 7.15783746714623122624908908756, 8.171247902776691425934121477779, 8.414567196239562792229225112125, 9.569539170935395360804162049378, 10.29162586724221102452789282898

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