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Properties

Label 2-768-16.11-c2-0-22
Degree $2$
Conductor $768$
Sign $0.382 + 0.923i$
Analytic cond. $20.9264$
Root an. cond. $4.57454$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.22 + 1.22i)3-s + (0.732 + 0.732i)5-s − 11.9·7-s + 2.99i·9-s + (14.4 − 14.4i)11-s + (−7.39 + 7.39i)13-s + 1.79i·15-s − 5.60·17-s + (−11.9 − 11.9i)19-s + (−14.6 − 14.6i)21-s + 41.4·23-s − 23.9i·25-s + (−3.67 + 3.67i)27-s + (−4.19 + 4.19i)29-s − 20.2i·31-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (0.146 + 0.146i)5-s − 1.71·7-s + 0.333i·9-s + (1.31 − 1.31i)11-s + (−0.568 + 0.568i)13-s + 0.119i·15-s − 0.329·17-s + (−0.630 − 0.630i)19-s + (−0.698 − 0.698i)21-s + 1.80·23-s − 0.957i·25-s + (−0.136 + 0.136i)27-s + (−0.144 + 0.144i)29-s − 0.653i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.382 + 0.923i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.382 + 0.923i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.368142790\)
\(L(\frac12)\) \(\approx\) \(1.368142790\)
\(L(2)\) 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 \( 1 + (-1.22 - 1.22i)T \)
good5 \( 1 + (-0.732 - 0.732i)T + 25iT^{2} \)
7 \( 1 + 11.9T + 49T^{2} \)
11 \( 1 + (-14.4 + 14.4i)T - 121iT^{2} \)
13 \( 1 + (7.39 - 7.39i)T - 169iT^{2} \)
17 \( 1 + 5.60T + 289T^{2} \)
19 \( 1 + (11.9 + 11.9i)T + 361iT^{2} \)
23 \( 1 - 41.4T + 529T^{2} \)
29 \( 1 + (4.19 - 4.19i)T - 841iT^{2} \)
31 \( 1 + 20.2iT - 961T^{2} \)
37 \( 1 + (17.9 + 17.9i)T + 1.36e3iT^{2} \)
41 \( 1 + 39.1iT - 1.68e3T^{2} \)
43 \( 1 + (-29.2 + 29.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 56.5iT - 2.20e3T^{2} \)
53 \( 1 + (7.51 + 7.51i)T + 2.80e3iT^{2} \)
59 \( 1 + (-33.5 + 33.5i)T - 3.48e3iT^{2} \)
61 \( 1 + (39.4 - 39.4i)T - 3.72e3iT^{2} \)
67 \( 1 + (-38.2 - 38.2i)T + 4.48e3iT^{2} \)
71 \( 1 - 26.9T + 5.04e3T^{2} \)
73 \( 1 - 85.5iT - 5.32e3T^{2} \)
79 \( 1 + 66.8iT - 6.24e3T^{2} \)
83 \( 1 + (102. + 102. i)T + 6.88e3iT^{2} \)
89 \( 1 - 85.2iT - 7.92e3T^{2} \)
97 \( 1 - 33.0T + 9.40e3T^{2} \)
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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

−9.820963226191527371371636779676, −8.966105912323669704230339049355, −8.797067784128512316765830780616, −7.02823668670179145344643364669, −6.61842919316674708516877316515, −5.62946934727022035345727161443, −4.19814180116400213618905204606, −3.41505475800381138396001425815, −2.47438690974719181404070146193, −0.46740448400418919776047641457, 1.31451437237726367297885667637, 2.72676010516123833479319999301, 3.63200686718285352978371672235, 4.80023122375334564016198607965, 6.18392078186020694962413232051, 6.81381736415934307795926610273, 7.46825370567866852913816320537, 8.825475430868281598618098574351, 9.469340624587567626504672553103, 9.923965751331932331818416429382

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