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Properties

Label 2-768-16.3-c2-0-14
Degree $2$
Conductor $768$
Sign $0.923 - 0.382i$
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

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

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(20.9264\)
Root analytic conductor: \(4.57454\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{768} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 768,\ (\ :1),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.670434778\)
\(L(\frac12)\) \(\approx\) \(2.670434778\)
\(L(2)\) 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

−10.10959351856876700676846768094, −9.195657643528462810725582435055, −8.363771079374398396435151241276, −7.67607429624683265817141531651, −6.84282682816096471873650481688, −5.84516803356639325568281250189, −4.41546555444845313060124636915, −3.99496594726903060339968580876, −2.04444056085766914233526512421, −1.52641521482766079946368893711, 0.985124290894519211003397592755, 2.28514666350276142950855244271, 3.79767406460877607158050370255, 4.40679186680962510759538403867, 5.55592480521878352275537574938, 6.47104185552827415151318342635, 7.84990775259606453264469428048, 8.484915381022390486592287671143, 8.858769252455269831765012848798, 10.16317967035928860574635858708

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