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Properties

Label 1-13e2-169.43-r0-0-0
Degree $1$
Conductor $169$
Sign $-0.181 - 0.983i$
Analytic cond. $0.784832$
Root an. cond. $0.784832$
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  + (0.200 − 0.979i)2-s + (0.987 − 0.160i)3-s + (−0.919 − 0.391i)4-s + (0.970 + 0.239i)5-s + (0.0402 − 0.999i)6-s + (−0.428 − 0.903i)7-s + (−0.568 + 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (−0.948 − 0.316i)11-s + (−0.970 − 0.239i)12-s + (−0.970 + 0.239i)14-s + (0.996 + 0.0804i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.120 − 0.992i)18-s + ⋯
L(s)  = 1  + (0.200 − 0.979i)2-s + (0.987 − 0.160i)3-s + (−0.919 − 0.391i)4-s + (0.970 + 0.239i)5-s + (0.0402 − 0.999i)6-s + (−0.428 − 0.903i)7-s + (−0.568 + 0.822i)8-s + (0.948 − 0.316i)9-s + (0.428 − 0.903i)10-s + (−0.948 − 0.316i)11-s + (−0.970 − 0.239i)12-s + (−0.970 + 0.239i)14-s + (0.996 + 0.0804i)15-s + (0.692 + 0.721i)16-s + (0.428 + 0.903i)17-s + (−0.120 − 0.992i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(169\)    =    \(13^{2}\)
Sign: $-0.181 - 0.983i$
Analytic conductor: \(0.784832\)
Root analytic conductor: \(0.784832\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{169} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 169,\ (0:\ ),\ -0.181 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.062803255 - 1.277278199i\)
\(L(\frac12)\) \(\approx\) \(1.062803255 - 1.277278199i\)
\(L(1)\) \(\approx\) \(1.219693978 - 0.8686215854i\)
\(L(1)\) \(\approx\) \(1.219693978 - 0.8686215854i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
good2 \( 1 + (0.200 - 0.979i)T \)
3 \( 1 + (0.987 - 0.160i)T \)
5 \( 1 + (0.970 + 0.239i)T \)
7 \( 1 + (-0.428 - 0.903i)T \)
11 \( 1 + (-0.948 - 0.316i)T \)
17 \( 1 + (0.428 + 0.903i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.200 + 0.979i)T \)
31 \( 1 + (-0.885 + 0.464i)T \)
37 \( 1 + (0.845 + 0.534i)T \)
41 \( 1 + (-0.987 + 0.160i)T \)
43 \( 1 + (-0.845 + 0.534i)T \)
47 \( 1 + (-0.120 + 0.992i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (-0.692 + 0.721i)T \)
61 \( 1 + (-0.996 + 0.0804i)T \)
67 \( 1 + (0.919 - 0.391i)T \)
71 \( 1 + (0.632 + 0.774i)T \)
73 \( 1 + (0.748 - 0.663i)T \)
79 \( 1 + (0.120 - 0.992i)T \)
83 \( 1 + (0.354 + 0.935i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.278 + 0.960i)T \)
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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

−27.60231516057998360393954653699, −26.48877165916521532321404697764, −25.65234041377456314455075945742, −25.14872220145419031571033910611, −24.43419062330036711938604214680, −23.115717700042912657055233795612, −21.91894661530608209698490196229, −21.28577032846314578075390880881, −20.26171384224575701201082109421, −18.521355841511062349742008094873, −18.33399678855779141129982338927, −16.757449886856285175930071875736, −15.81567510620216113014836587638, −15.034483239678013074862669980439, −13.9278360040404141155603355919, −13.26323844598732868799123383490, −12.28451379335134549734633315061, −9.89012385276588746782017672091, −9.47854162228245550627054188373, −8.32222192571184584402619135339, −7.33589652194084465662605561957, −5.8625189855648994287219832284, −5.05625431128930705745328866333, −3.433034326381149436010398006, −2.17690016099572181953701858759, 1.387466249789245342976251334342, 2.66856096725923485307582611851, 3.53753168760157029866309357786, 4.987836910306621917094559664258, 6.51871653447389976830153509205, 7.993294576027509366836283509734, 9.20637050181841924257800126342, 10.13516207298148809939080015994, 10.78333225009465260462011074618, 12.66785223034238733556989561507, 13.29405453004478864018310328303, 14.01899750188457363615676879016, 14.93581893884601109011611114812, 16.558523778309130006497777155270, 17.96539367738988439840672776154, 18.616229138297409882005409052462, 19.775731535927897944237257480668, 20.42395040790569610030655983369, 21.38017213684642330364495282230, 22.08617996907876675849132910954, 23.47128363496170716134701122615, 24.25846717643866989934225359776, 25.85952595697823253055964653777, 26.19467048687067675990501524423, 27.20319790169046858202741975537

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