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Properties

Label 2-768-24.5-c2-0-35
Degree $2$
Conductor $768$
Sign $0.762 + 0.646i$
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  + (−2.98 + 0.246i)3-s + 6.63·5-s + 0.578·7-s + (8.87 − 1.47i)9-s + 8.68·11-s − 17.9i·13-s + (−19.8 + 1.63i)15-s − 19.0i·17-s + 32.1i·19-s + (−1.72 + 0.142i)21-s − 20.4i·23-s + 19.0·25-s + (−26.1 + 6.59i)27-s + 22.0·29-s − 26.2·31-s + ⋯
L(s)  = 1  + (−0.996 + 0.0821i)3-s + 1.32·5-s + 0.0825·7-s + (0.986 − 0.163i)9-s + 0.789·11-s − 1.37i·13-s + (−1.32 + 0.109i)15-s − 1.11i·17-s + 1.69i·19-s + (−0.0823 + 0.00678i)21-s − 0.890i·23-s + 0.761·25-s + (−0.969 + 0.244i)27-s + 0.759·29-s − 0.846·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.762 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.762 + 0.646i$
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: \((2,\ 768,\ (\ :1),\ 0.762 + 0.646i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.771862407\)
\(L(\frac12)\) \(\approx\) \(1.771862407\)
\(L(2)\) 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 \( 1 + (2.98 - 0.246i)T \)
good5 \( 1 - 6.63T + 25T^{2} \)
7 \( 1 - 0.578T + 49T^{2} \)
11 \( 1 - 8.68T + 121T^{2} \)
13 \( 1 + 17.9iT - 169T^{2} \)
17 \( 1 + 19.0iT - 289T^{2} \)
19 \( 1 - 32.1iT - 361T^{2} \)
23 \( 1 + 20.4iT - 529T^{2} \)
29 \( 1 - 22.0T + 841T^{2} \)
31 \( 1 + 26.2T + 961T^{2} \)
37 \( 1 - 53.3iT - 1.36e3T^{2} \)
41 \( 1 + 35.6iT - 1.68e3T^{2} \)
43 \( 1 + 50.4iT - 1.84e3T^{2} \)
47 \( 1 + 30.6iT - 2.20e3T^{2} \)
53 \( 1 - 88.8T + 2.80e3T^{2} \)
59 \( 1 - 63.1T + 3.48e3T^{2} \)
61 \( 1 - 33.5iT - 3.72e3T^{2} \)
67 \( 1 + 108. iT - 4.48e3T^{2} \)
71 \( 1 + 59.3iT - 5.04e3T^{2} \)
73 \( 1 - 5.60T + 5.32e3T^{2} \)
79 \( 1 - 78.9T + 6.24e3T^{2} \)
83 \( 1 + 48.5T + 6.88e3T^{2} \)
89 \( 1 - 58.7iT - 7.92e3T^{2} \)
97 \( 1 - 93.3T + 9.40e3T^{2} \)
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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

−10.21860864733113222191079845536, −9.465313323311000200722636406766, −8.381283800857325956189830423101, −7.17884392240866723086922429981, −6.29335174336229765593566377194, −5.62732517546250622400809466939, −4.92795257149108007216085960231, −3.54730605244091642934283730002, −2.02140536178473440145989083306, −0.792855697823626883550754282544, 1.24765990110437050170836693269, 2.16631905985778602257032904381, 4.00846756405678028331668969317, 4.95224583437778820459324957558, 5.93277960569191451569403701740, 6.52870595318263124997534826416, 7.26056140060962445932595142859, 8.830504409025987580857977176887, 9.462373207339395805792257463728, 10.16860931038363695744637343968

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