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Properties

Label 2-768-24.11-c1-0-21
Degree $2$
Conductor $768$
Sign $0.912 + 0.408i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.61 − 0.618i)3-s + 3.23·5-s − 1.23i·7-s + (2.23 − 2.00i)9-s + 5.23i·11-s − 4.47i·13-s + (5.23 − 2.00i)15-s + 2.47i·17-s − 0.763·19-s + (−0.763 − 2.00i)21-s − 2.47·23-s + 5.47·25-s + (2.38 − 4.61i)27-s − 4.76·29-s + 5.23i·31-s + ⋯
L(s)  = 1  + (0.934 − 0.356i)3-s + 1.44·5-s − 0.467i·7-s + (0.745 − 0.666i)9-s + 1.57i·11-s − 1.24i·13-s + (1.35 − 0.516i)15-s + 0.599i·17-s − 0.175·19-s + (−0.166 − 0.436i)21-s − 0.515·23-s + 1.09·25-s + (0.458 − 0.888i)27-s − 0.884·29-s + 0.940i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.59004 - 0.552772i\)
\(L(\frac12)\) \(\approx\) \(2.59004 - 0.552772i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 0.618i)T \)
good5 \( 1 - 3.23T + 5T^{2} \)
7 \( 1 + 1.23iT - 7T^{2} \)
11 \( 1 - 5.23iT - 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 - 2.47iT - 17T^{2} \)
19 \( 1 + 0.763T + 19T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + 4.76T + 29T^{2} \)
31 \( 1 - 5.23iT - 31T^{2} \)
37 \( 1 + 8.47iT - 37T^{2} \)
41 \( 1 - 6.47iT - 41T^{2} \)
43 \( 1 + 7.23T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 3.23T + 53T^{2} \)
59 \( 1 + 1.23iT - 59T^{2} \)
61 \( 1 - 0.472iT - 61T^{2} \)
67 \( 1 + 9.70T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 0.291iT - 79T^{2} \)
83 \( 1 + 2.76iT - 83T^{2} \)
89 \( 1 - 4iT - 89T^{2} \)
97 \( 1 - 0.472T + 97T^{2} \)
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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

−9.967747750130581379896288161707, −9.642205906359530606507862295212, −8.583887245706870611453429293850, −7.65997510621501046073561142931, −6.89368360211312632214411479199, −5.94214237878264546495717936384, −4.85552592127439032413977388274, −3.61323731348774623382985820408, −2.34510526927726976772639631533, −1.55750505639739302578898936595, 1.74680060758197382820664442667, 2.63835358966363840630741313498, 3.76120586643914427108095633574, 5.05925155461602201975485784058, 5.94180968628171064633908720927, 6.78043772433461745121400803666, 8.093752874611012393459303381108, 8.894342667010724727090506826060, 9.407653812544736019728636391114, 10.10500470436477351429724884799

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