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Properties

Label 2-768-16.11-c2-0-24
Degree $2$
Conductor $768$
Sign $0.991 + 0.130i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.22 − 1.22i)3-s + (5.27 + 5.27i)5-s + 12.7·7-s + 2.99i·9-s + (7.46 − 7.46i)11-s + (11.9 − 11.9i)13-s − 12.9i·15-s − 4.55·17-s + (−14.6 − 14.6i)19-s + (−15.6 − 15.6i)21-s + 31.6·23-s + 30.7i·25-s + (3.67 − 3.67i)27-s + (−38.3 + 38.3i)29-s + 13.8i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (1.05 + 1.05i)5-s + 1.82·7-s + 0.333i·9-s + (0.678 − 0.678i)11-s + (0.919 − 0.919i)13-s − 0.862i·15-s − 0.268·17-s + (−0.769 − 0.769i)19-s + (−0.743 − 0.743i)21-s + 1.37·23-s + 1.22i·25-s + (0.136 − 0.136i)27-s + (−1.32 + 1.32i)29-s + 0.447i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.991 + 0.130i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.617033004\)
\(L(\frac12)\) \(\approx\) \(2.617033004\)
\(L(2)\) 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 \( 1 + (1.22 + 1.22i)T \)
good5 \( 1 + (-5.27 - 5.27i)T + 25iT^{2} \)
7 \( 1 - 12.7T + 49T^{2} \)
11 \( 1 + (-7.46 + 7.46i)T - 121iT^{2} \)
13 \( 1 + (-11.9 + 11.9i)T - 169iT^{2} \)
17 \( 1 + 4.55T + 289T^{2} \)
19 \( 1 + (14.6 + 14.6i)T + 361iT^{2} \)
23 \( 1 - 31.6T + 529T^{2} \)
29 \( 1 + (38.3 - 38.3i)T - 841iT^{2} \)
31 \( 1 - 13.8iT - 961T^{2} \)
37 \( 1 + (28.7 + 28.7i)T + 1.36e3iT^{2} \)
41 \( 1 - 3.98iT - 1.68e3T^{2} \)
43 \( 1 + (-10.7 + 10.7i)T - 1.84e3iT^{2} \)
47 \( 1 + 27.1iT - 2.20e3T^{2} \)
53 \( 1 + (43.3 + 43.3i)T + 2.80e3iT^{2} \)
59 \( 1 + (6.84 - 6.84i)T - 3.48e3iT^{2} \)
61 \( 1 + (8.18 - 8.18i)T - 3.72e3iT^{2} \)
67 \( 1 + (26.6 + 26.6i)T + 4.48e3iT^{2} \)
71 \( 1 - 16.7T + 5.04e3T^{2} \)
73 \( 1 + 50.1iT - 5.32e3T^{2} \)
79 \( 1 - 75.2iT - 6.24e3T^{2} \)
83 \( 1 + (-0.382 - 0.382i)T + 6.88e3iT^{2} \)
89 \( 1 - 165. iT - 7.92e3T^{2} \)
97 \( 1 + 63.7T + 9.40e3T^{2} \)
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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

−10.57997960412308751009026842656, −9.073053043123980087484232913443, −8.468460256789984519096239342755, −7.33582255476356584168202796305, −6.60699368278224691261278381961, −5.66832986226123505095033654884, −4.99624398027126096919651536029, −3.45204243317300901380902936286, −2.14410935715367295814750711230, −1.19651737205926045643933918148, 1.36083551177570459599169328208, 1.88285969151171046495413167450, 4.17992598777688732689804881971, 4.65974861690693546063323406960, 5.56840435800281883287978960755, 6.38312216282412026105574373506, 7.67562135572929586647577867184, 8.760822279166569850635229549287, 9.113949750587792110539439330204, 10.09557692903785501160148503515

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