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Properties

Label 2-768-32.13-c1-0-4
Degree $2$
Conductor $768$
Sign $0.874 - 0.484i$
Analytic cond. $6.13251$
Root an. cond. $2.47639$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.382 − 0.923i)3-s + (1.20 − 0.498i)5-s + (−2.59 + 2.59i)7-s + (−0.707 − 0.707i)9-s + (2.14 + 5.18i)11-s + (0.984 + 0.407i)13-s − 1.30i·15-s − 0.979i·17-s + (5.68 + 2.35i)19-s + (1.40 + 3.38i)21-s + (3.70 + 3.70i)23-s + (−2.33 + 2.33i)25-s + (−0.923 + 0.382i)27-s + (1.17 − 2.83i)29-s − 1.54·31-s + ⋯
L(s)  = 1  + (0.220 − 0.533i)3-s + (0.538 − 0.223i)5-s + (−0.980 + 0.980i)7-s + (−0.235 − 0.235i)9-s + (0.647 + 1.56i)11-s + (0.272 + 0.113i)13-s − 0.336i·15-s − 0.237i·17-s + (1.30 + 0.540i)19-s + (0.306 + 0.739i)21-s + (0.771 + 0.771i)23-s + (−0.466 + 0.466i)25-s + (−0.177 + 0.0736i)27-s + (0.217 − 0.525i)29-s − 0.277·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.874 - 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.874 - 0.484i$
Analytic conductor: \(6.13251\)
Root analytic conductor: \(2.47639\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1/2),\ 0.874 - 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.60756 + 0.415181i\)
\(L(\frac12)\) \(\approx\) \(1.60756 + 0.415181i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.382 + 0.923i)T \)
good5 \( 1 + (-1.20 + 0.498i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (2.59 - 2.59i)T - 7iT^{2} \)
11 \( 1 + (-2.14 - 5.18i)T + (-7.77 + 7.77i)T^{2} \)
13 \( 1 + (-0.984 - 0.407i)T + (9.19 + 9.19i)T^{2} \)
17 \( 1 + 0.979iT - 17T^{2} \)
19 \( 1 + (-5.68 - 2.35i)T + (13.4 + 13.4i)T^{2} \)
23 \( 1 + (-3.70 - 3.70i)T + 23iT^{2} \)
29 \( 1 + (-1.17 + 2.83i)T + (-20.5 - 20.5i)T^{2} \)
31 \( 1 + 1.54T + 31T^{2} \)
37 \( 1 + (-8.23 + 3.41i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (1.10 + 1.10i)T + 41iT^{2} \)
43 \( 1 + (3.47 + 8.37i)T + (-30.4 + 30.4i)T^{2} \)
47 \( 1 + 3.15iT - 47T^{2} \)
53 \( 1 + (-2.55 - 6.16i)T + (-37.4 + 37.4i)T^{2} \)
59 \( 1 + (8.95 - 3.70i)T + (41.7 - 41.7i)T^{2} \)
61 \( 1 + (-2.00 + 4.84i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 + (-1.14 + 2.76i)T + (-47.3 - 47.3i)T^{2} \)
71 \( 1 + (10.0 - 10.0i)T - 71iT^{2} \)
73 \( 1 + (-8.11 - 8.11i)T + 73iT^{2} \)
79 \( 1 + 0.155iT - 79T^{2} \)
83 \( 1 + (-5.13 - 2.12i)T + (58.6 + 58.6i)T^{2} \)
89 \( 1 + (6.15 - 6.15i)T - 89iT^{2} \)
97 \( 1 - 14.3T + 97T^{2} \)
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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

−10.00443073503849275597162497512, −9.440279251444748283491856719621, −9.003947777078059555258778813196, −7.65033039807063477671726674324, −6.94523448538922314347844775273, −5.99206015906429112385709508214, −5.24695794671012816040549289872, −3.79006248680779092340416994366, −2.60586790736940920083620240423, −1.53018226241745634410052254126, 0.905020079589747679044407099532, 2.98624462140885191767345571239, 3.52673651108563857219830640291, 4.74001406891346711376924402291, 6.03986845794946355982000513028, 6.53734808389614110548256501920, 7.69991510521935403922064972527, 8.754758448462947897424474928800, 9.487306074763271122728047805712, 10.18564419454532575413882121433

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