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Properties

Label 2-768-8.3-c2-0-28
Degree $2$
Conductor $768$
Sign $-0.707 + 0.707i$
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  + 1.73·3-s − 8.29i·5-s + 8.55i·7-s + 2.99·9-s − 13.7·11-s − 17.0i·13-s − 14.3i·15-s + 20.3·17-s − 20.4·19-s + 14.8i·21-s − 5.51i·23-s − 43.7·25-s + 5.19·27-s − 41.0i·29-s − 22.2i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.65i·5-s + 1.22i·7-s + 0.333·9-s − 1.25·11-s − 1.31i·13-s − 0.957i·15-s + 1.19·17-s − 1.07·19-s + 0.705i·21-s − 0.239i·23-s − 1.75·25-s + 0.192·27-s − 1.41i·29-s − 0.719i·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.378833739\)
\(L(\frac12)\) \(\approx\) \(1.378833739\)
\(L(2)\) 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 \( 1 - 1.73T \)
good5 \( 1 + 8.29iT - 25T^{2} \)
7 \( 1 - 8.55iT - 49T^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + 17.0iT - 169T^{2} \)
17 \( 1 - 20.3T + 289T^{2} \)
19 \( 1 + 20.4T + 361T^{2} \)
23 \( 1 + 5.51iT - 529T^{2} \)
29 \( 1 + 41.0iT - 841T^{2} \)
31 \( 1 + 22.2iT - 961T^{2} \)
37 \( 1 + 11.6iT - 1.36e3T^{2} \)
41 \( 1 + 35.9T + 1.68e3T^{2} \)
43 \( 1 + 66.8T + 1.84e3T^{2} \)
47 \( 1 + 19.9iT - 2.20e3T^{2} \)
53 \( 1 - 17.6iT - 2.80e3T^{2} \)
59 \( 1 - 62.1T + 3.48e3T^{2} \)
61 \( 1 - 47.4iT - 3.72e3T^{2} \)
67 \( 1 + 74.8T + 4.48e3T^{2} \)
71 \( 1 - 16.9iT - 5.04e3T^{2} \)
73 \( 1 + 101.T + 5.32e3T^{2} \)
79 \( 1 + 0.879iT - 6.24e3T^{2} \)
83 \( 1 + 23.2T + 6.88e3T^{2} \)
89 \( 1 - 16.3T + 7.92e3T^{2} \)
97 \( 1 - 188.T + 9.40e3T^{2} \)
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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

−9.761255470080911800344661679444, −8.628243424104615777459522834921, −8.357509030408561115449108024342, −7.65278123718299511267131695278, −5.82682495049922364284074055507, −5.40802379311347482321485809030, −4.44657767185931172115229143800, −3.03116198551327179687102577316, −1.98357872382588314576154740825, −0.41218694707554599475652862946, 1.79827414733399350043363559959, 3.04129628365632032452494666049, 3.70055109856584286681831633949, 4.92177714226048055694228599419, 6.41712952517970104355530659109, 7.08025969313383412841286008551, 7.63161446836318334576620688186, 8.629996577025402539556976419053, 9.967490842028339995466801694205, 10.36464735234778904118624312645

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