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Properties

Label 1-161-161.40-r0-0-0
Degree $1$
Conductor $161$
Sign $0.839 + 0.542i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.839 + 0.542i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (40, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.839 + 0.542i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045719920 + 0.3086228147i\)
\(L(\frac12)\) \(\approx\) \(1.045719920 + 0.3086228147i\)
\(L(1)\) \(\approx\) \(1.003176607 + 0.2591866426i\)
\(L(1)\) \(\approx\) \(1.003176607 + 0.2591866426i\)

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

−27.5225625812778233593988671703, −26.97258153184166909053882394024, −26.13855076348703640516896307613, −25.0254159208067022344131189409, −23.68011397285161533820742011396, −22.59679592278744835407243193641, −21.528975289228156631272127069305, −20.684456363692809391784139371152, −19.89858312682945622707801978766, −19.031413794662466466899510575973, −18.372325273006452258769874150005, −16.52365459987937675699324322839, −15.97600303133136892580393071841, −14.41560330015396306591928655781, −13.66930936003627990301755653205, −12.3548246628171515806674424767, −11.32937333366467470867306341788, −10.394727480330788193876597936630, −9.06915486971826494570106482061, −8.42555687752972460032179469749, −7.39015082721629514616151594012, −5.07267914415036294479222142678, −3.609908645025310443227134231393, −3.30031599446087417613949509219, −1.362376306543031833559420215034, 1.28288459267915074440240765188, 3.37169480278422472134940546689, 4.484077665305426953315049194913, 6.212955166721858589908936595117, 7.40636967347974527174735425741, 7.964785651547817640950889767891, 9.045056186061157594487947663740, 10.1308777409095152621071514553, 11.81455603417925004369411730068, 13.02825355992816266260884505319, 14.06124264387194165859930842408, 15.13839445409172426258962476928, 15.57678210484492198154458072556, 16.91572403596141317455918705068, 18.19330781671525124572889602842, 18.857712394907029891906649986151, 19.82865384229550724870447079771, 20.66944567037843881465352658675, 22.46854408710652839883549876741, 23.33863909312058674443728527201, 24.08729074742724681921779007236, 25.11304491009279007788531379859, 25.86174115656771518222541372457, 26.66952664135249706122006534649, 27.65811171972684452228788972077

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