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Properties

Label 1-2e4-16.13-r0-0-0
Degree $1$
Conductor $16$
Sign $0.923 + 0.382i$
Analytic cond. $0.0743036$
Root an. cond. $0.0743036$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s − i·5-s − 7-s − 9-s − i·11-s + i·13-s + 15-s + 17-s + i·19-s − i·21-s − 23-s − 25-s − i·27-s + i·29-s + 31-s + ⋯
L(s)  = 1  + i·3-s − i·5-s − 7-s − 9-s − i·11-s + i·13-s + 15-s + 17-s + i·19-s − i·21-s − 23-s − 25-s − i·27-s + i·29-s + 31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(16\)    =    \(2^{4}\)
Sign: $0.923 + 0.382i$
Analytic conductor: \(0.0743036\)
Root analytic conductor: \(0.0743036\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{16} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 16,\ (0:\ ),\ 0.923 + 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5686330845 + 0.1131081530i\)
\(L(\frac12)\) \(\approx\) \(0.5686330845 + 0.1131081530i\)
\(L(1)\) \(\approx\) \(0.8231312585 + 0.1227420153i\)
\(L(1)\) \(\approx\) \(0.8231312585 + 0.1227420153i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + T \)
5 \( 1 \)
7 \( 1 + iT \)
11 \( 1 \)
13 \( 1 - iT \)
17 \( 1 \)
19 \( 1 - T \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - iT \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 + T \)
59 \( 1 \)
61 \( 1 + T \)
67 \( 1 \)
71 \( 1 + iT \)
73 \( 1 \)
79 \( 1 - iT \)
83 \( 1 \)
89 \( 1 - T \)
97 \( 1 \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−41.825082929984968853222680180957, −41.11799879193078332101458852009, −39.25392742897261456429460065312, −37.90608073784334260891653490262, −36.401486661873487004385414016400, −35.16404762115227997892341380688, −34.08416262895376380849995774529, −32.22231906459055023095942343526, −30.54702187745248757785934145189, −29.759002793480986369420867910563, −28.23877796729846247962053513839, −26.11408114872351890342704573396, −25.22485016834369120152191524053, −23.290769667113505204788611690537, −22.39155264073640549163110845185, −19.97014430214759194559391690124, −18.74512871871518406028546682032, −17.48207262781754948392319550715, −15.24798500104019565503155333391, −13.55309931536034432804826020610, −12.08914445233257224925151296912, −10.07040957004959561723835012593, −7.600203099156647857273129151538, −6.24222661642667706187762514208, −2.84244102407636072925895428429, 3.77219663606750472419782799253, 5.728047333435860519362213721584, 8.66786640849238501209889646485, 10.00259023243655542290253810714, 12.049942431652227153052593142170, 13.97991629550442986927238238690, 16.08548800464759590948198175057, 16.681722732861615769456343264957, 19.23493382126471425806593111566, 20.72796697239924433369945403917, 21.92754860270152894860021415533, 23.587559115072946395276902944601, 25.38693819074342021651702639106, 26.77854143732071190127066260033, 28.17945954776589941181522637086, 29.268268469038841948778888603080, 31.73447436171577321987099122908, 32.2767795639539696932046786377, 33.72936878893931858442913481399, 35.381559992056068894043463063433, 36.77823002794198262805113689625, 38.32433311383906940121148924132, 39.36311167696596713658796752847, 40.53989990047074163425905900710, 42.453640260382987390856937027248

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