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Properties

Label 1-161-161.25-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.673 + 0.739i$
Analytic cond. $0.747680$
Root an. cond. $0.747680$
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  + (−0.995 − 0.0950i)2-s + (0.723 + 0.690i)3-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (−0.654 − 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)10-s + (−0.995 + 0.0950i)11-s + (0.580 + 0.814i)12-s + (−0.142 + 0.989i)13-s + (−0.959 − 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.0475 − 0.998i)18-s + (−0.327 − 0.945i)19-s + ⋯
L(s)  = 1  + (−0.995 − 0.0950i)2-s + (0.723 + 0.690i)3-s + (0.981 + 0.189i)4-s + (−0.888 + 0.458i)5-s + (−0.654 − 0.755i)6-s + (−0.959 − 0.281i)8-s + (0.0475 + 0.998i)9-s + (0.928 − 0.371i)10-s + (−0.995 + 0.0950i)11-s + (0.580 + 0.814i)12-s + (−0.142 + 0.989i)13-s + (−0.959 − 0.281i)15-s + (0.928 + 0.371i)16-s + (−0.327 + 0.945i)17-s + (0.0475 − 0.998i)18-s + (−0.327 − 0.945i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.673 + 0.739i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.673 + 0.739i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.673 + 0.739i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2276910305 + 0.5153285787i\)
\(L(\frac12)\) \(\approx\) \(0.2276910305 + 0.5153285787i\)
\(L(1)\) \(\approx\) \(0.5754219779 + 0.3022701172i\)
\(L(1)\) \(\approx\) \(0.5754219779 + 0.3022701172i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.995 - 0.0950i)T \)
3 \( 1 + (0.723 + 0.690i)T \)
5 \( 1 + (-0.888 + 0.458i)T \)
11 \( 1 + (-0.995 + 0.0950i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (-0.327 + 0.945i)T \)
19 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (0.0475 + 0.998i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.959 + 0.281i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.786 + 0.618i)T \)
59 \( 1 + (0.928 - 0.371i)T \)
61 \( 1 + (0.723 - 0.690i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.415 + 0.909i)T \)
73 \( 1 + (0.981 + 0.189i)T \)
79 \( 1 + (-0.786 - 0.618i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (0.235 - 0.971i)T \)
97 \( 1 + (0.841 + 0.540i)T \)
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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

−27.26053310399388400767331397622, −26.554228212226913407594288476248, −25.50062980878040932521562338302, −24.67894558317677032741495882914, −23.92279348066426012609617627291, −22.93336967753786427326024414891, −20.86728728767756635082442122030, −20.38251620228060249977795687791, −19.46882315435972780583992166283, −18.594739353356708020882382886089, −17.85264824505936933258569889895, −16.475729448587538087681023713760, −15.55188590281973938576142711753, −14.71356741389118293281658306934, −13.1019529411767905504510358112, −12.24054556620984602807206218765, −11.07741651368460333591901802240, −9.76382837991757308092433220716, −8.55605430782077237667808659870, −7.88086145895113573310091215235, −7.10573064445569711962817179349, −5.50063238710751086828457673839, −3.4776937410693504491493881196, −2.250746328389813754480299171075, −0.56465430443335467013385693869, 2.18105309592107526581832564178, 3.279688779871956033258259402, 4.59394853096851288725493908165, 6.62204421024071234591777798510, 7.78711272938321695022523617030, 8.52524229830868100410655568780, 9.67697016391919805406998240142, 10.71304153313654466053900381617, 11.44751929605088709960629403822, 12.95925611452580147687835000430, 14.54944841512894038660927777213, 15.44389651156865712492651863829, 16.0399104749742304652626478468, 17.22193773903309204002444630907, 18.603875995114933223653987795124, 19.313153599954112382173412980621, 20.057554363730994295549301221, 21.13545379413322969178563957980, 21.91868433661538504553562899756, 23.50848096200445817060166162021, 24.41877886409288415824508191787, 25.83364861176628360476858037419, 26.31456253869383774845410474376, 26.94505022456109379645696983431, 28.032038081971127013651490580244

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