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Properties

Label 2-768-24.11-c3-0-31
Degree $2$
Conductor $768$
Sign $0.383 - 0.923i$
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

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

Invariants

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

Particular Values

\(L(2)\) \(\approx\) \(1.592305999\)
\(L(\frac12)\) \(\approx\) \(1.592305999\)
\(L(\frac{5}{2})\) 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

−10.03350102233437683075182107664, −9.529769299241894724943708233243, −8.163071845280303469890600829751, −7.62871307457615517967571799658, −6.96914291019791017436236372069, −5.15862701780866053673474088444, −4.61917462499503298646885239898, −3.65679449476118595826920459418, −2.85091596234192445406053172474, −0.856499148492855464390948390948, 0.57113392962517758285710394361, 1.98420245944748182024804946065, 3.14395516576421969730972359968, 3.98659385593583424613768162508, 5.60706867949213955582481596400, 6.22655527866209740748596818431, 7.31155866327833637968079557989, 8.122591281658027865479148257442, 8.781451783718168258234415864701, 9.319858977558989134230009555230

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