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Properties

Label 2-768-24.11-c3-0-18
Degree $2$
Conductor $768$
Sign $-0.130 - 0.991i$
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  + (−3.16 + 4.12i)3-s − 21.4·5-s + 20.9i·7-s + (−6.98 − 26.0i)9-s − 9.94i·11-s − 67.8i·13-s + (67.7 − 88.2i)15-s − 7.97i·17-s + 62.4·19-s + (−86.1 − 66.1i)21-s + 101.·23-s + 333.·25-s + (129. + 53.7i)27-s − 122.·29-s + 87.5i·31-s + ⋯
L(s)  = 1  + (−0.608 + 0.793i)3-s − 1.91·5-s + 1.12i·7-s + (−0.258 − 0.965i)9-s − 0.272i·11-s − 1.44i·13-s + (1.16 − 1.51i)15-s − 0.113i·17-s + 0.753·19-s + (−0.895 − 0.687i)21-s + 0.923·23-s + 2.66·25-s + (0.923 + 0.383i)27-s − 0.784·29-s + 0.506i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.130 - 0.991i$
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: \((2,\ 768,\ (\ :3/2),\ -0.130 - 0.991i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6362303386\)
\(L(\frac12)\) \(\approx\) \(0.6362303386\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.16 - 4.12i)T \)
good5 \( 1 + 21.4T + 125T^{2} \)
7 \( 1 - 20.9iT - 343T^{2} \)
11 \( 1 + 9.94iT - 1.33e3T^{2} \)
13 \( 1 + 67.8iT - 2.19e3T^{2} \)
17 \( 1 + 7.97iT - 4.91e3T^{2} \)
19 \( 1 - 62.4T + 6.85e3T^{2} \)
23 \( 1 - 101.T + 1.21e4T^{2} \)
29 \( 1 + 122.T + 2.43e4T^{2} \)
31 \( 1 - 87.5iT - 2.97e4T^{2} \)
37 \( 1 + 106. iT - 5.06e4T^{2} \)
41 \( 1 + 90.3iT - 6.89e4T^{2} \)
43 \( 1 + 451.T + 7.95e4T^{2} \)
47 \( 1 + 428.T + 1.03e5T^{2} \)
53 \( 1 + 362.T + 1.48e5T^{2} \)
59 \( 1 + 801. iT - 2.05e5T^{2} \)
61 \( 1 - 647. iT - 2.26e5T^{2} \)
67 \( 1 - 957.T + 3.00e5T^{2} \)
71 \( 1 + 224.T + 3.57e5T^{2} \)
73 \( 1 - 108.T + 3.89e5T^{2} \)
79 \( 1 + 615. iT - 4.93e5T^{2} \)
83 \( 1 - 204. iT - 5.71e5T^{2} \)
89 \( 1 - 454. iT - 7.04e5T^{2} \)
97 \( 1 - 740.T + 9.12e5T^{2} \)
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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

−10.32882635608651623412100862229, −9.222265665889259280141985972934, −8.453938693711798052522193452118, −7.73738083220034523005813813184, −6.65426427355131295008983905600, −5.38897720456118153828209597814, −4.91473552945807565938598573908, −3.55699013182815955787450879152, −3.10917305900557411697522810713, −0.66013051290382755389853240902, 0.34412818870646686364193092822, 1.46550010621752422759969034222, 3.29261091532326924608877415509, 4.26831917650244263293183927749, 4.95936646390842192112318457655, 6.63727156016991680735145754164, 7.11378304525267014992927027388, 7.73226024706101291850548384373, 8.519879799864641156778413742019, 9.788849502198959105350153667399

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