Properties

Label 2-768-24.5-c2-0-43
Degree $2$
Conductor $768$
Sign $0.465 + 0.884i$
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  + (−0.888 + 2.86i)3-s + 8.59·5-s − 10.9·7-s + (−7.41 − 5.09i)9-s − 2.75·11-s − 4.43i·13-s + (−7.63 + 24.6i)15-s − 25.4i·17-s − 17.5i·19-s + (9.71 − 31.3i)21-s − 17.5i·23-s + 48.8·25-s + (21.1 − 16.7i)27-s − 19.6·29-s + 2.58·31-s + ⋯
L(s)  = 1  + (−0.296 + 0.955i)3-s + 1.71·5-s − 1.56·7-s + (−0.824 − 0.565i)9-s − 0.250·11-s − 0.341i·13-s + (−0.509 + 1.64i)15-s − 1.49i·17-s − 0.923i·19-s + (0.462 − 1.49i)21-s − 0.762i·23-s + 1.95·25-s + (0.784 − 0.619i)27-s − 0.676·29-s + 0.0833·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.465 + 0.884i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.465 + 0.884i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.465 + 0.884i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.203535435\)
\(L(\frac12)\) \(\approx\) \(1.203535435\)
\(L(2)\) 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 \( 1 + (0.888 - 2.86i)T \)
good5 \( 1 - 8.59T + 25T^{2} \)
7 \( 1 + 10.9T + 49T^{2} \)
11 \( 1 + 2.75T + 121T^{2} \)
13 \( 1 + 4.43iT - 169T^{2} \)
17 \( 1 + 25.4iT - 289T^{2} \)
19 \( 1 + 17.5iT - 361T^{2} \)
23 \( 1 + 17.5iT - 529T^{2} \)
29 \( 1 + 19.6T + 841T^{2} \)
31 \( 1 - 2.58T + 961T^{2} \)
37 \( 1 + 7.73iT - 1.36e3T^{2} \)
41 \( 1 - 58.0iT - 1.68e3T^{2} \)
43 \( 1 + 42.1iT - 1.84e3T^{2} \)
47 \( 1 + 17.4iT - 2.20e3T^{2} \)
53 \( 1 + 69.0T + 2.80e3T^{2} \)
59 \( 1 - 50.5T + 3.48e3T^{2} \)
61 \( 1 + 32.5iT - 3.72e3T^{2} \)
67 \( 1 - 48.0iT - 4.48e3T^{2} \)
71 \( 1 + 22.1iT - 5.04e3T^{2} \)
73 \( 1 - 27.0T + 5.32e3T^{2} \)
79 \( 1 + 97.4T + 6.24e3T^{2} \)
83 \( 1 - 59.5T + 6.88e3T^{2} \)
89 \( 1 + 110. iT - 7.92e3T^{2} \)
97 \( 1 - 55.1T + 9.40e3T^{2} \)
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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

−9.805571004079482305430800479718, −9.485247136800635825592530154041, −8.761528846661484617197215585607, −7.00428100284637375200980708797, −6.28530510990315241615641326445, −5.54410441786180828799786319437, −4.74139997358377429763355522373, −3.20160052275411699667505855038, −2.53278279409610997093452861520, −0.41160100102280899906025713768, 1.43675760688999368902591272299, 2.34934969621310695104976107935, 3.54315137116872522351199113491, 5.37289176363192832290506740477, 6.13972640192688604175295048402, 6.39047949906435463989147857493, 7.49633442627853053179841236602, 8.677185563570024691098125250444, 9.543311021086079621690379479288, 10.15392700032618885624644228647

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