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Properties

Label 2-768-8.3-c4-0-39
Degree $2$
Conductor $768$
Sign $0.707 + 0.707i$
Analytic cond. $79.3881$
Root an. cond. $8.91000$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.19·3-s − 8.12i·5-s + 42.5i·7-s + 27·9-s − 166.·11-s − 38.3i·13-s − 42.2i·15-s + 44.5·17-s − 191.·19-s + 220. i·21-s − 122. i·23-s + 558.·25-s + 140.·27-s + 132. i·29-s − 1.22e3i·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.324i·5-s + 0.867i·7-s + 0.333·9-s − 1.37·11-s − 0.226i·13-s − 0.187i·15-s + 0.154·17-s − 0.531·19-s + 0.500i·21-s − 0.232i·23-s + 0.894·25-s + 0.192·27-s + 0.157i·29-s − 1.27i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(79.3881\)
Root analytic conductor: \(8.91000\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :2),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.140959679\)
\(L(\frac12)\) \(\approx\) \(2.140959679\)
\(L(3)\) 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 \( 1 - 5.19T \)
good5 \( 1 + 8.12iT - 625T^{2} \)
7 \( 1 - 42.5iT - 2.40e3T^{2} \)
11 \( 1 + 166.T + 1.46e4T^{2} \)
13 \( 1 + 38.3iT - 2.85e4T^{2} \)
17 \( 1 - 44.5T + 8.35e4T^{2} \)
19 \( 1 + 191.T + 1.30e5T^{2} \)
23 \( 1 + 122. iT - 2.79e5T^{2} \)
29 \( 1 - 132. iT - 7.07e5T^{2} \)
31 \( 1 + 1.22e3iT - 9.23e5T^{2} \)
37 \( 1 + 1.29e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.49e3T + 2.82e6T^{2} \)
43 \( 1 - 1.58e3T + 3.41e6T^{2} \)
47 \( 1 + 2.22e3iT - 4.87e6T^{2} \)
53 \( 1 - 4.57e3iT - 7.89e6T^{2} \)
59 \( 1 - 4.54e3T + 1.21e7T^{2} \)
61 \( 1 + 4.99e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.11e3T + 2.01e7T^{2} \)
71 \( 1 - 4.22e3iT - 2.54e7T^{2} \)
73 \( 1 - 6.98e3T + 2.83e7T^{2} \)
79 \( 1 + 4.12e3iT - 3.89e7T^{2} \)
83 \( 1 + 6.86e3T + 4.74e7T^{2} \)
89 \( 1 - 8.14e3T + 6.27e7T^{2} \)
97 \( 1 + 1.15e4T + 8.85e7T^{2} \)
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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

−9.494661517427539254669978552212, −8.733715094507906147531181649008, −8.060359993848317976634862481026, −7.24336565201726086048604411972, −5.93992486124872674032973577499, −5.21703321331854917180786178133, −4.14435926598887756784661613126, −2.81335224755338699026048669708, −2.17336705472534760760771579933, −0.53872277515808256485002947222, 0.932771406859790502662589544800, 2.34975543083396869778306015001, 3.24878611584210685726202696814, 4.32597724729667379209777036136, 5.27848004776541994610566944019, 6.56917551968131891139437257524, 7.35080870498712177417953943554, 8.067595459994822877288928028327, 8.921703717643263802650840933316, 10.03681694487394356836087736973

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