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Properties

Label 2-768-24.11-c3-0-58
Degree $2$
Conductor $768$
Sign $0.872 + 0.489i$
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  + (1.40 + 5.00i)3-s − 5.86·5-s + 5.92i·7-s + (−23.0 + 14.0i)9-s − 27.9i·11-s + 0.0653i·13-s + (−8.26 − 29.3i)15-s − 36.9i·17-s − 30.7·19-s + (−29.6 + 8.33i)21-s + 61.2·23-s − 90.5·25-s + (−102. − 95.3i)27-s + 143.·29-s − 299. i·31-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)3-s − 0.524·5-s + 0.319i·7-s + (−0.853 + 0.521i)9-s − 0.766i·11-s + 0.00139i·13-s + (−0.142 − 0.505i)15-s − 0.527i·17-s − 0.371·19-s + (−0.307 + 0.0866i)21-s + 0.555·23-s − 0.724·25-s + (−0.733 − 0.679i)27-s + 0.919·29-s − 1.73i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.872 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.872 + 0.489i$
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: \((2,\ 768,\ (\ :3/2),\ 0.872 + 0.489i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.339043821\)
\(L(\frac12)\) \(\approx\) \(1.339043821\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.40 - 5.00i)T \)
good5 \( 1 + 5.86T + 125T^{2} \)
7 \( 1 - 5.92iT - 343T^{2} \)
11 \( 1 + 27.9iT - 1.33e3T^{2} \)
13 \( 1 - 0.0653iT - 2.19e3T^{2} \)
17 \( 1 + 36.9iT - 4.91e3T^{2} \)
19 \( 1 + 30.7T + 6.85e3T^{2} \)
23 \( 1 - 61.2T + 1.21e4T^{2} \)
29 \( 1 - 143.T + 2.43e4T^{2} \)
31 \( 1 + 299. iT - 2.97e4T^{2} \)
37 \( 1 - 340. iT - 5.06e4T^{2} \)
41 \( 1 + 379. iT - 6.89e4T^{2} \)
43 \( 1 + 470.T + 7.95e4T^{2} \)
47 \( 1 - 428.T + 1.03e5T^{2} \)
53 \( 1 - 505.T + 1.48e5T^{2} \)
59 \( 1 - 207. iT - 2.05e5T^{2} \)
61 \( 1 + 578. iT - 2.26e5T^{2} \)
67 \( 1 - 415.T + 3.00e5T^{2} \)
71 \( 1 + 547.T + 3.57e5T^{2} \)
73 \( 1 + 194.T + 3.89e5T^{2} \)
79 \( 1 + 308. iT - 4.93e5T^{2} \)
83 \( 1 - 62.3iT - 5.71e5T^{2} \)
89 \( 1 + 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 - 703.T + 9.12e5T^{2} \)
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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

−9.898066945664489703739193438870, −8.914256740052248318609727230783, −8.393286135379245928992996911268, −7.44294508322199302179505697659, −6.16995234445353488934741400658, −5.27474785371369990980265460877, −4.29276674242740744598387558001, −3.41599621117182798265589655372, −2.40242153525916308463142986930, −0.40842835052708751638223993882, 1.01793623683243787119632690300, 2.19466570617549422883651518042, 3.39933566344466362262889440398, 4.45910013118631210564481786758, 5.69503082844584288776055255030, 6.81033559246366350306738083195, 7.29989618515613630587339686397, 8.247549123437470114050885016467, 8.878939254007444513900070052769, 10.03986297697939511448316421113

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