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Properties

Label 2-768-16.11-c2-0-12
Degree $2$
Conductor $768$
Sign $0.991 + 0.130i$
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 + (−4.54 − 4.54i)5-s − 1.15·7-s + 2.99i·9-s + (−6.42 + 6.42i)11-s + (−14.8 + 14.8i)13-s + 11.1i·15-s + 15.0·17-s + (9.44 + 9.44i)19-s + (1.41 + 1.41i)21-s + 31.6·23-s + 16.3i·25-s + (3.67 − 3.67i)27-s + (5.51 − 5.51i)29-s − 20.3i·31-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.909 − 0.909i)5-s − 0.164·7-s + 0.333i·9-s + (−0.584 + 0.584i)11-s + (−1.14 + 1.14i)13-s + 0.742i·15-s + 0.887·17-s + (0.497 + 0.497i)19-s + (0.0671 + 0.0671i)21-s + 1.37·23-s + 0.653i·25-s + (0.136 − 0.136i)27-s + (0.190 − 0.190i)29-s − 0.656i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(768\)    =    \(2^{8} \cdot 3\)
Sign: $0.991 + 0.130i$
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: \((2,\ 768,\ (\ :1),\ 0.991 + 0.130i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9918010140\)
\(L(\frac12)\) \(\approx\) \(0.9918010140\)
\(L(2)\) 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 \( 1 + (4.54 + 4.54i)T + 25iT^{2} \)
7 \( 1 + 1.15T + 49T^{2} \)
11 \( 1 + (6.42 - 6.42i)T - 121iT^{2} \)
13 \( 1 + (14.8 - 14.8i)T - 169iT^{2} \)
17 \( 1 - 15.0T + 289T^{2} \)
19 \( 1 + (-9.44 - 9.44i)T + 361iT^{2} \)
23 \( 1 - 31.6T + 529T^{2} \)
29 \( 1 + (-5.51 + 5.51i)T - 841iT^{2} \)
31 \( 1 + 20.3iT - 961T^{2} \)
37 \( 1 + (50.3 + 50.3i)T + 1.36e3iT^{2} \)
41 \( 1 - 52.4iT - 1.68e3T^{2} \)
43 \( 1 + (-42.2 + 42.2i)T - 1.84e3iT^{2} \)
47 \( 1 + 27.1iT - 2.20e3T^{2} \)
53 \( 1 + (-20.1 - 20.1i)T + 2.80e3iT^{2} \)
59 \( 1 + (-69.0 + 69.0i)T - 3.48e3iT^{2} \)
61 \( 1 + (0.992 - 0.992i)T - 3.72e3iT^{2} \)
67 \( 1 + (-77.0 - 77.0i)T + 4.48e3iT^{2} \)
71 \( 1 - 44.5T + 5.04e3T^{2} \)
73 \( 1 - 3.56iT - 5.32e3T^{2} \)
79 \( 1 - 33.5iT - 6.24e3T^{2} \)
83 \( 1 + (-90.2 - 90.2i)T + 6.88e3iT^{2} \)
89 \( 1 - 68.3iT - 7.92e3T^{2} \)
97 \( 1 - 161.T + 9.40e3T^{2} \)
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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

−10.00659254511575009121907711408, −9.285091576585217918228808555312, −8.246686271978905769141403525792, −7.46778525996510428846031773343, −6.86578024788488817371076267125, −5.40273317593187669791514615873, −4.82365293898242229833404553783, −3.76386408708631357780990352836, −2.21688943085660190732740324151, −0.75424077854937566497575714581, 0.57159451261384116810293567982, 2.97317661331514099642339588622, 3.32361728498179344191145298056, 4.86634285272079039157558756870, 5.51038616782717510729464036250, 6.82206317524476297527090226225, 7.46455032798956233495577605086, 8.284031556581103260892065678233, 9.462253453918568371679860376451, 10.42352013734029462602178594346

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