Properties

Label 2-768-24.11-c3-0-36
Degree $2$
Conductor $768$
Sign $0.707 - 0.707i$
Analytic cond. $45.3134$
Root an. cond. $6.73152$
Motivic weight $3$
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 + 31.1i·7-s + 27·9-s − 70i·13-s + 155.·19-s − 162i·21-s − 125·25-s − 140.·27-s − 155. i·31-s + 110i·37-s + 363. i·39-s + 218.·43-s − 629·49-s − 810·57-s − 182i·61-s + ⋯
L(s)  = 1  − 1.00·3-s + 1.68i·7-s + 9-s − 1.49i·13-s + 1.88·19-s − 1.68i·21-s − 25-s − 1.00·27-s − 0.903i·31-s + 0.488i·37-s + 1.49i·39-s + 0.773·43-s − 1.83·49-s − 1.88·57-s − 0.382i·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(45.3134\)
Root analytic conductor: \(6.73152\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :3/2),\ 0.707 - 0.707i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.362973739\)
\(L(\frac12)\) \(\approx\) \(1.362973739\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.19T \)
good5 \( 1 + 125T^{2} \)
7 \( 1 - 31.1iT - 343T^{2} \)
11 \( 1 - 1.33e3T^{2} \)
13 \( 1 + 70iT - 2.19e3T^{2} \)
17 \( 1 - 4.91e3T^{2} \)
19 \( 1 - 155.T + 6.85e3T^{2} \)
23 \( 1 + 1.21e4T^{2} \)
29 \( 1 + 2.43e4T^{2} \)
31 \( 1 + 155. iT - 2.97e4T^{2} \)
37 \( 1 - 110iT - 5.06e4T^{2} \)
41 \( 1 - 6.89e4T^{2} \)
43 \( 1 - 218.T + 7.95e4T^{2} \)
47 \( 1 + 1.03e5T^{2} \)
53 \( 1 + 1.48e5T^{2} \)
59 \( 1 - 2.05e5T^{2} \)
61 \( 1 + 182iT - 2.26e5T^{2} \)
67 \( 1 - 654.T + 3.00e5T^{2} \)
71 \( 1 + 3.57e5T^{2} \)
73 \( 1 - 1.19e3T + 3.89e5T^{2} \)
79 \( 1 - 1.09e3iT - 4.93e5T^{2} \)
83 \( 1 - 5.71e5T^{2} \)
89 \( 1 - 7.04e5T^{2} \)
97 \( 1 - 1.33e3T + 9.12e5T^{2} \)
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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

−9.906629284311014729182702365414, −9.457723474860444949751843307843, −8.216285344807150643698356668974, −7.49186686124235496714134414860, −6.19682136607361985120193838625, −5.55893651135371698107494499026, −5.06377479949282502051850852729, −3.49052771749545545092349896134, −2.31416284767667679521119821656, −0.817882817300581082198919422250, 0.63050624707714154825009934729, 1.62137991434244596865235610125, 3.62047601524181520318039335250, 4.36144709228684008276385914395, 5.28635984110818305702164819532, 6.44854651293924181005811722430, 7.16467530793755123608534981311, 7.71603898980661197534846933213, 9.289799993430142439616095631090, 9.925950123392018401974744948104

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