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Properties

Label 1-161-161.132-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.764 - 0.644i$
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  + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (0.654 − 0.755i)12-s + (0.959 − 0.281i)13-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + (0.415 − 0.909i)18-s + (−0.142 + 0.989i)19-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (0.415 − 0.909i)5-s + (0.142 + 0.989i)6-s + (0.841 − 0.540i)8-s + (0.415 + 0.909i)9-s + (−0.959 + 0.281i)10-s + (0.654 − 0.755i)11-s + (0.654 − 0.755i)12-s + (0.959 − 0.281i)13-s + (−0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + (0.415 − 0.909i)18-s + (−0.142 + 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.764 - 0.644i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.764 - 0.644i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2180324695 - 0.5966399603i\)
\(L(\frac12)\) \(\approx\) \(0.2180324695 - 0.5966399603i\)
\(L(1)\) \(\approx\) \(0.4986271423 - 0.4347501709i\)
\(L(1)\) \(\approx\) \(0.4986271423 - 0.4347501709i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + (-0.841 - 0.540i)T \)
5 \( 1 + (0.415 - 0.909i)T \)
11 \( 1 + (0.654 - 0.755i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (-0.142 - 0.989i)T \)
19 \( 1 + (-0.142 + 0.989i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.841 + 0.540i)T \)
37 \( 1 + (-0.415 - 0.909i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (-0.841 - 0.540i)T \)
47 \( 1 - T \)
53 \( 1 + (0.959 + 0.281i)T \)
59 \( 1 + (0.959 - 0.281i)T \)
61 \( 1 + (0.841 - 0.540i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.142 - 0.989i)T \)
79 \( 1 + (0.959 - 0.281i)T \)
83 \( 1 + (0.415 + 0.909i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
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

−27.9892900028774248450203406688, −27.25560394149928335789126798967, −25.98193699697998832301990384861, −25.79251632915681651094925382655, −24.20825922769926711632978129349, −23.33378689915079100041124594065, −22.473322369514329938518938687998, −21.638085326164475092612940421356, −20.22334856974294646621588764962, −18.934096929878654785599243021359, −17.96602515724918005584047893381, −17.37280531895168709356018734976, −16.37559821991271631249676705263, −15.23498972075297812141573070050, −14.66487554684451763101486538361, −13.2445100684186460594795402825, −11.48718822776663156375670719651, −10.6711253127042574272760713899, −9.79214500060649557684495590592, −8.75392032157110504982822932056, −6.97127832985629699023758208718, −6.44377376719076425739155483117, −5.2932657327746562605205797104, −3.88576218001370271057452721725, −1.64871255878768995867177756000, 0.802866696272522296463450144293, 1.88634489203604345665169947988, 3.77566703716867311987496177939, 5.25568326158553464494039557899, 6.47408877675984108838068349951, 7.95083910928779547301472746357, 8.90087031056658487446545686986, 10.09573315013369146658078979317, 11.263454122043851061987087253584, 12.0204995228708000408241993803, 13.05330135058468614934910858605, 13.79244872037721486880691126122, 16.18605138015820998535351231195, 16.60321628825596102455021783879, 17.67770388522213544950904151899, 18.4286122664025951375335889298, 19.439753706000747548209577755969, 20.54392813363357877649857240136, 21.40775516826510842155590746309, 22.41617829856523798837157228761, 23.42799393407636405063552909855, 24.822519445899349920743594757786, 25.19623428275171171283767811782, 26.82728767262370656827077291882, 27.73859367160972513141268030689

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