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Properties

Label 2-768-8.3-c6-0-30
Degree $2$
Conductor $768$
Sign $0.707 - 0.707i$
Analytic cond. $176.681$
Root an. cond. $13.2921$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 15.5·3-s − 95.8i·5-s − 655. i·7-s + 243·9-s + 1.00e3·11-s + 3.44e3i·13-s − 1.49e3i·15-s + 4.52e3·17-s − 8.67e3·19-s − 1.02e4i·21-s + 2.28e4i·23-s + 6.43e3·25-s + 3.78e3·27-s − 1.90e3i·29-s + 3.65e4i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.766i·5-s − 1.91i·7-s + 0.333·9-s + 0.751·11-s + 1.56i·13-s − 0.442i·15-s + 0.920·17-s − 1.26·19-s − 1.10i·21-s + 1.88i·23-s + 0.412·25-s + 0.192·27-s − 0.0780i·29-s + 1.22i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(176.681\)
Root analytic conductor: \(13.2921\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.342276118\)
\(L(\frac12)\) \(\approx\) \(2.342276118\)
\(L(4)\) 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 \( 1 - 15.5T \)
good5 \( 1 + 95.8iT - 1.56e4T^{2} \)
7 \( 1 + 655. iT - 1.17e5T^{2} \)
11 \( 1 - 1.00e3T + 1.77e6T^{2} \)
13 \( 1 - 3.44e3iT - 4.82e6T^{2} \)
17 \( 1 - 4.52e3T + 2.41e7T^{2} \)
19 \( 1 + 8.67e3T + 4.70e7T^{2} \)
23 \( 1 - 2.28e4iT - 1.48e8T^{2} \)
29 \( 1 + 1.90e3iT - 5.94e8T^{2} \)
31 \( 1 - 3.65e4iT - 8.87e8T^{2} \)
37 \( 1 - 1.91e4iT - 2.56e9T^{2} \)
41 \( 1 - 3.05e4T + 4.75e9T^{2} \)
43 \( 1 - 1.88e4T + 6.32e9T^{2} \)
47 \( 1 - 1.53e5iT - 1.07e10T^{2} \)
53 \( 1 - 2.60e5iT - 2.21e10T^{2} \)
59 \( 1 + 2.41e5T + 4.21e10T^{2} \)
61 \( 1 - 2.04e5iT - 5.15e10T^{2} \)
67 \( 1 + 4.21e5T + 9.04e10T^{2} \)
71 \( 1 - 4.34e5iT - 1.28e11T^{2} \)
73 \( 1 + 3.85e5T + 1.51e11T^{2} \)
79 \( 1 + 1.18e5iT - 2.43e11T^{2} \)
83 \( 1 - 4.12e5T + 3.26e11T^{2} \)
89 \( 1 - 3.41e5T + 4.96e11T^{2} \)
97 \( 1 - 1.08e6T + 8.32e11T^{2} \)
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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

−9.360553822404512632068634861086, −8.811529120486506478811226420723, −7.64863113462736062338213700757, −7.13790304411895812318463728446, −6.16892319738620952927407248187, −4.58239315784544556353472976237, −4.20172511132611285489748352831, −3.27515519421935200288239466344, −1.46355137253919734970097167584, −1.20358109136167555104310625298, 0.39632480895635355809719087696, 2.05254608305103729979064715813, 2.69566094103152008191798936961, 3.49714624342011958936296640881, 4.87727300734553613848821172189, 5.94816429928518915460536759014, 6.48929983848595401941259419762, 7.79565458668223596487114987794, 8.497450627216837711093253496558, 9.113178944610677653929814118258

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