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Properties

Label 2-768-24.11-c3-0-90
Degree $2$
Conductor $768$
Sign $-0.923 + 0.383i$
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  + (1.98 − 4.80i)3-s + 11.9·5-s − 22.6i·7-s + (−19.1 − 19.0i)9-s − 61.9i·11-s − 71.3i·13-s + (23.7 − 57.5i)15-s + 74.3i·17-s + 108.·19-s + (−109. − 44.9i)21-s − 24.7·23-s + 18.6·25-s + (−129. + 54.2i)27-s − 84.1·29-s + 130. i·31-s + ⋯
L(s)  = 1  + (0.381 − 0.924i)3-s + 1.07·5-s − 1.22i·7-s + (−0.709 − 0.705i)9-s − 1.69i·11-s − 1.52i·13-s + (0.408 − 0.990i)15-s + 1.06i·17-s + 1.30·19-s + (−1.13 − 0.467i)21-s − 0.223·23-s + 0.149·25-s + (−0.922 + 0.386i)27-s − 0.538·29-s + 0.755i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.923 + 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $-0.923 + 0.383i$
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: \((2,\ 768,\ (\ :3/2),\ -0.923 + 0.383i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.599884465\)
\(L(\frac12)\) \(\approx\) \(2.599884465\)
\(L(\frac{5}{2})\) 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 \( 1 + (-1.98 + 4.80i)T \)
good5 \( 1 - 11.9T + 125T^{2} \)
7 \( 1 + 22.6iT - 343T^{2} \)
11 \( 1 + 61.9iT - 1.33e3T^{2} \)
13 \( 1 + 71.3iT - 2.19e3T^{2} \)
17 \( 1 - 74.3iT - 4.91e3T^{2} \)
19 \( 1 - 108.T + 6.85e3T^{2} \)
23 \( 1 + 24.7T + 1.21e4T^{2} \)
29 \( 1 + 84.1T + 2.43e4T^{2} \)
31 \( 1 - 130. iT - 2.97e4T^{2} \)
37 \( 1 - 397. iT - 5.06e4T^{2} \)
41 \( 1 + 104. iT - 6.89e4T^{2} \)
43 \( 1 - 161.T + 7.95e4T^{2} \)
47 \( 1 + 72.8T + 1.03e5T^{2} \)
53 \( 1 - 136.T + 1.48e5T^{2} \)
59 \( 1 - 243. iT - 2.05e5T^{2} \)
61 \( 1 - 358. iT - 2.26e5T^{2} \)
67 \( 1 - 449.T + 3.00e5T^{2} \)
71 \( 1 - 329.T + 3.57e5T^{2} \)
73 \( 1 - 925.T + 3.89e5T^{2} \)
79 \( 1 - 55.9iT - 4.93e5T^{2} \)
83 \( 1 + 928. iT - 5.71e5T^{2} \)
89 \( 1 + 853. iT - 7.04e5T^{2} \)
97 \( 1 + 714.T + 9.12e5T^{2} \)
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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

−9.602381616474142935821507449496, −8.449360747185404684219018777811, −7.938026958439139069454267368395, −6.95025432083474602171925990856, −5.97308038535555640826567169216, −5.47845306760449225856473016391, −3.62224268000212913097913184946, −2.92871512128120288732655525649, −1.40429049119403704666644521318, −0.66250930201020315470080546072, 2.01728170919183389778791043237, 2.42993514379645490191187167260, 3.98182289179852226590575762014, 5.03560994861533415562493715996, 5.58579675868527440676561786327, 6.77815568059291735121872437687, 7.78806683043966768468703915200, 9.178033208353178128364524381993, 9.468723505323530459607271201357, 9.766242554253557495846698931350

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