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Properties

Label 2-768-16.3-c2-0-21
Degree $2$
Conductor $768$
Sign $-0.130 + 0.991i$
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 + (−3.71 + 3.71i)5-s − 5.63·7-s − 2.99i·9-s + (5.26 + 5.26i)11-s + (−1.74 − 1.74i)13-s + 9.11i·15-s − 1.43·17-s + (22.1 − 22.1i)19-s + (−6.90 + 6.90i)21-s − 2.27·23-s − 2.67i·25-s + (−3.67 − 3.67i)27-s + (−35.9 − 35.9i)29-s − 13.3i·31-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.743 + 0.743i)5-s − 0.805·7-s − 0.333i·9-s + (0.478 + 0.478i)11-s + (−0.133 − 0.133i)13-s + 0.607i·15-s − 0.0846·17-s + (1.16 − 1.16i)19-s + (−0.328 + 0.328i)21-s − 0.0988·23-s − 0.106i·25-s + (−0.136 − 0.136i)27-s + (−1.23 − 1.23i)29-s − 0.429i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.130 + 0.991i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.130 + 0.991i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.108142267\)
\(L(\frac12)\) \(\approx\) \(1.108142267\)
\(L(2)\) 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 \( 1 + (3.71 - 3.71i)T - 25iT^{2} \)
7 \( 1 + 5.63T + 49T^{2} \)
11 \( 1 + (-5.26 - 5.26i)T + 121iT^{2} \)
13 \( 1 + (1.74 + 1.74i)T + 169iT^{2} \)
17 \( 1 + 1.43T + 289T^{2} \)
19 \( 1 + (-22.1 + 22.1i)T - 361iT^{2} \)
23 \( 1 + 2.27T + 529T^{2} \)
29 \( 1 + (35.9 + 35.9i)T + 841iT^{2} \)
31 \( 1 + 13.3iT - 961T^{2} \)
37 \( 1 + (-36.1 + 36.1i)T - 1.36e3iT^{2} \)
41 \( 1 + 63.2iT - 1.68e3T^{2} \)
43 \( 1 + (46.5 + 46.5i)T + 1.84e3iT^{2} \)
47 \( 1 - 61.0iT - 2.20e3T^{2} \)
53 \( 1 + (-49.0 + 49.0i)T - 2.80e3iT^{2} \)
59 \( 1 + (-41.6 - 41.6i)T + 3.48e3iT^{2} \)
61 \( 1 + (40.4 + 40.4i)T + 3.72e3iT^{2} \)
67 \( 1 + (-36.0 + 36.0i)T - 4.48e3iT^{2} \)
71 \( 1 - 8.24T + 5.04e3T^{2} \)
73 \( 1 + 122. iT - 5.32e3T^{2} \)
79 \( 1 - 93.0iT - 6.24e3T^{2} \)
83 \( 1 + (-21.6 + 21.6i)T - 6.88e3iT^{2} \)
89 \( 1 - 119. iT - 7.92e3T^{2} \)
97 \( 1 + 115.T + 9.40e3T^{2} \)
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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

−9.632181215007680127730984579798, −9.252928814100859298426659853402, −7.943315358836796676758392746386, −7.24937429045329309424411461960, −6.70332577738029003862381100865, −5.54431292352321551894414424136, −4.07403336381606802700473530665, −3.30113589915089394444296524349, −2.25182306419685054093174443774, −0.39018200297421202012403544070, 1.27051183719230011023459309062, 3.10790873810493648026079826302, 3.79238726567923521410438893122, 4.81993051508227964560275382800, 5.86097705429613116639179096869, 6.96909258728244142164284134944, 7.990334142183319709246122075852, 8.626902857517836441462810081495, 9.536203903302006472479244336199, 10.07052769043194348004524443491

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