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Properties

Label 1-161-161.4-r0-0-0
Degree $1$
Conductor $161$
Sign $0.947 - 0.320i$
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  + (−0.327 − 0.945i)2-s + (−0.888 − 0.458i)3-s + (−0.786 + 0.618i)4-s + (−0.995 − 0.0950i)5-s + (−0.142 + 0.989i)6-s + (0.841 + 0.540i)8-s + (0.580 + 0.814i)9-s + (0.235 + 0.971i)10-s + (−0.327 + 0.945i)11-s + (0.981 − 0.189i)12-s + (−0.959 − 0.281i)13-s + (0.841 + 0.540i)15-s + (0.235 − 0.971i)16-s + (0.928 − 0.371i)17-s + (0.580 − 0.814i)18-s + (0.928 + 0.371i)19-s + ⋯
L(s)  = 1  + (−0.327 − 0.945i)2-s + (−0.888 − 0.458i)3-s + (−0.786 + 0.618i)4-s + (−0.995 − 0.0950i)5-s + (−0.142 + 0.989i)6-s + (0.841 + 0.540i)8-s + (0.580 + 0.814i)9-s + (0.235 + 0.971i)10-s + (−0.327 + 0.945i)11-s + (0.981 − 0.189i)12-s + (−0.959 − 0.281i)13-s + (0.841 + 0.540i)15-s + (0.235 − 0.971i)16-s + (0.928 − 0.371i)17-s + (0.580 − 0.814i)18-s + (0.928 + 0.371i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.947 - 0.320i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4444831322 - 0.07319836231i\)
\(L(\frac12)\) \(\approx\) \(0.4444831322 - 0.07319836231i\)
\(L(1)\) \(\approx\) \(0.4967947139 - 0.1921229643i\)
\(L(1)\) \(\approx\) \(0.4967947139 - 0.1921229643i\)

Euler product

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

−27.63710881360641665026030290144, −26.77524949613548987758512674497, −26.36109355873267439799663802905, −24.64183623106670727507882055743, −23.94459736831697007433785713767, −23.15178135993098855201508453842, −22.32353627872556585660932854201, −21.2923176298778364032690359626, −19.62803569899240331965058563471, −18.803744486563328137461081402, −17.75684196328991547021852372852, −16.65545733385219860905948354478, −16.08178023740117306037701770501, −15.18637905233816162378147777200, −14.15857054175797566898839184069, −12.58221926258976903933862358748, −11.47517670762567687325409445057, −10.41691345891664362134696635107, −9.310879967515698871336084487992, −7.9660724061808513790139088894, −7.041723003446637288857639330135, −5.76637047016295462509932961899, −4.815040324187256546266633097557, −3.59443369598516828505014631397, −0.6053853140338135203592392313, 1.109935004023501036979564997170, 2.772024575296615499754179228056, 4.34445074952192864022662650591, 5.28987913647980880926743852165, 7.35950554948446423204960233685, 7.81315177669100327625292076589, 9.58278185863060822690038150367, 10.536691018880181169385448456758, 11.72833319551072955152938581395, 12.21487527546427143648477470605, 13.096402659909153645292198599064, 14.65570482297697393036096294119, 16.136221322142946668820806674218, 17.026957867608233042319041511113, 18.10035229843927305014668984656, 18.82178975745682898306417088448, 19.83898529462474806466253360894, 20.66914868722347446214601506888, 22.08047677149673690226567060513, 22.7889058978076153394886914493, 23.5190208966736030717846695857, 24.68993099276760627238517162360, 26.061467909523699261379053377190, 27.37589884226006928968075711963, 27.66839484607074519406763638923

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