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Properties

Label 2-1344-56.27-c3-0-60
Degree $2$
Conductor $1344$
Sign $0.775 - 0.631i$
Analytic cond. $79.2985$
Root an. cond. $8.90497$
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  + 3i·3-s + 15.6·5-s + (10.8 − 15.0i)7-s − 9·9-s + 41.3·11-s + 8.02·13-s + 47.0i·15-s + 14.2i·17-s − 10.2i·19-s + (45.0 + 32.5i)21-s + 203. i·23-s + 121.·25-s − 27i·27-s + 82.6i·29-s − 156.·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.40·5-s + (0.585 − 0.810i)7-s − 0.333·9-s + 1.13·11-s + 0.171·13-s + 0.810i·15-s + 0.202i·17-s − 0.123i·19-s + (0.468 + 0.337i)21-s + 1.84i·23-s + 0.969·25-s − 0.192i·27-s + 0.529i·29-s − 0.905·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1344\)    =    \(2^{6} \cdot 3 \cdot 7\)
Sign: $0.775 - 0.631i$
Analytic conductor: \(79.2985\)
Root analytic conductor: \(8.90497\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1344} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1344,\ (\ :3/2),\ 0.775 - 0.631i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.455911324\)
\(L(\frac12)\) \(\approx\) \(3.455911324\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
7 \( 1 + (-10.8 + 15.0i)T \)
good5 \( 1 - 15.6T + 125T^{2} \)
11 \( 1 - 41.3T + 1.33e3T^{2} \)
13 \( 1 - 8.02T + 2.19e3T^{2} \)
17 \( 1 - 14.2iT - 4.91e3T^{2} \)
19 \( 1 + 10.2iT - 6.85e3T^{2} \)
23 \( 1 - 203. iT - 1.21e4T^{2} \)
29 \( 1 - 82.6iT - 2.43e4T^{2} \)
31 \( 1 + 156.T + 2.97e4T^{2} \)
37 \( 1 - 106. iT - 5.06e4T^{2} \)
41 \( 1 - 315. iT - 6.89e4T^{2} \)
43 \( 1 - 154.T + 7.95e4T^{2} \)
47 \( 1 - 474.T + 1.03e5T^{2} \)
53 \( 1 - 266. iT - 1.48e5T^{2} \)
59 \( 1 + 425. iT - 2.05e5T^{2} \)
61 \( 1 + 8.30T + 2.26e5T^{2} \)
67 \( 1 - 820.T + 3.00e5T^{2} \)
71 \( 1 + 109. iT - 3.57e5T^{2} \)
73 \( 1 + 53.8iT - 3.89e5T^{2} \)
79 \( 1 + 1.17e3iT - 4.93e5T^{2} \)
83 \( 1 - 507. iT - 5.71e5T^{2} \)
89 \( 1 - 1.06e3iT - 7.04e5T^{2} \)
97 \( 1 + 573. iT - 9.12e5T^{2} \)
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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

−9.470877909358310317073919872148, −8.797922612878768122969447789111, −7.68207902386253784991162333508, −6.78141354991349141200987204997, −5.90635337307055899109080925437, −5.19868056949981225087347089802, −4.19559311432910441876427946319, −3.33958031795393944455269544911, −1.89210774623465044352414027704, −1.14549412639454458265780365998, 0.879066000493068619142196452798, 1.98141952726712361724509106553, 2.48489112628561117288005295703, 4.02328539691396421282545939052, 5.21331772371770665817725580246, 5.92907283005503446166512598467, 6.50347256136270227391385629043, 7.43225804641765267321087292721, 8.684266836044143522986122055369, 8.929929803400927974065846593209

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