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Properties

Label 1-161-161.144-r0-0-0
Degree $1$
Conductor $161$
Sign $0.661 - 0.749i$
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  + (0.981 − 0.189i)2-s + (0.0475 − 0.998i)3-s + (0.928 − 0.371i)4-s + (0.580 + 0.814i)5-s + (−0.142 − 0.989i)6-s + (0.841 − 0.540i)8-s + (−0.995 − 0.0950i)9-s + (0.723 + 0.690i)10-s + (0.981 + 0.189i)11-s + (−0.327 − 0.945i)12-s + (−0.959 + 0.281i)13-s + (0.841 − 0.540i)15-s + (0.723 − 0.690i)16-s + (−0.786 + 0.618i)17-s + (−0.995 + 0.0950i)18-s + (−0.786 − 0.618i)19-s + ⋯
L(s)  = 1  + (0.981 − 0.189i)2-s + (0.0475 − 0.998i)3-s + (0.928 − 0.371i)4-s + (0.580 + 0.814i)5-s + (−0.142 − 0.989i)6-s + (0.841 − 0.540i)8-s + (−0.995 − 0.0950i)9-s + (0.723 + 0.690i)10-s + (0.981 + 0.189i)11-s + (−0.327 − 0.945i)12-s + (−0.959 + 0.281i)13-s + (0.841 − 0.540i)15-s + (0.723 − 0.690i)16-s + (−0.786 + 0.618i)17-s + (−0.995 + 0.0950i)18-s + (−0.786 − 0.618i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.661 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.933486276 - 0.8721692594i\)
\(L(\frac12)\) \(\approx\) \(1.933486276 - 0.8721692594i\)
\(L(1)\) \(\approx\) \(1.797280772 - 0.5696957043i\)
\(L(1)\) \(\approx\) \(1.797280772 - 0.5696957043i\)

Euler product

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

−27.89411037268471230006069878244, −26.993362444612484156992274272082, −25.6894502103006386467921007449, −24.962223597244243267228630586269, −24.11818150940644978787551932623, −22.7799583369194901036641386572, −21.99770705515318207032232390890, −21.326132392500242043497285952967, −20.30277263118195363309719442471, −19.70722141868811176741474646596, −17.48095644738416070772609605326, −16.74125598477564743498309541093, −15.97257740958929613154701794167, −14.751032486503890538932253816445, −14.106717082852832822924905272909, −12.84708987218249928185045854719, −11.866809757658298356004056209617, −10.68432182784504855345620579791, −9.47054400444531769937055698089, −8.42808078625433723333649107186, −6.72229869778467760463240718497, −5.470230328733789400307964512218, −4.69901471864652853011192254909, −3.60134301430312549330528061077, −2.098622120483383042759689845760, 1.82289299808023309767681799349, 2.61443518058441655599220725938, 4.15200411726398852642564704166, 5.792787887547634677197755407003, 6.64046870482946080813562866015, 7.384660616288379070278961579301, 9.21473280958565211714165051480, 10.685272212308626629646918924608, 11.62866408333877899422641342574, 12.642641223061552563917305216303, 13.577039583087912898669937786426, 14.46131845481938756753099978870, 15.12112153183092999409125704963, 16.953424397160893449006308114186, 17.70517029933259148910153397811, 19.243819990333777725764891733549, 19.53816236016740256242980348826, 20.96276526882128764500537540339, 22.1733518850832312015778279371, 22.56623936666323128709395949395, 23.86520452984927308790920915771, 24.55107733910478631378313079417, 25.455892750168472406863535410356, 26.3068236195230945501989779365, 28.00358134702953467511903556098

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