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Properties

Label 1-161-161.87-r1-0-0
Degree $1$
Conductor $161$
Sign $0.845 - 0.534i$
Analytic cond. $17.3018$
Root an. cond. $17.3018$
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  + (−0.888 + 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (−0.723 + 0.690i)5-s + (−0.415 + 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)10-s + (−0.888 − 0.458i)11-s + (−0.0475 − 0.998i)12-s + (0.654 + 0.755i)13-s + (−0.142 + 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (0.235 + 0.971i)18-s + (0.995 + 0.0950i)19-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (0.786 − 0.618i)3-s + (0.580 − 0.814i)4-s + (−0.723 + 0.690i)5-s + (−0.415 + 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.235 − 0.971i)9-s + (0.327 − 0.945i)10-s + (−0.888 − 0.458i)11-s + (−0.0475 − 0.998i)12-s + (0.654 + 0.755i)13-s + (−0.142 + 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.995 − 0.0950i)17-s + (0.235 + 0.971i)18-s + (0.995 + 0.0950i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.845 - 0.534i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.845 - 0.534i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.264331812 - 0.3664248185i\)
\(L(\frac12)\) \(\approx\) \(1.264331812 - 0.3664248185i\)
\(L(1)\) \(\approx\) \(0.8873110046 - 0.04035346213i\)
\(L(1)\) \(\approx\) \(0.8873110046 - 0.04035346213i\)

Euler product

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

−27.70019234281398236935252980565, −26.81515355511777571939085234994, −25.83100788205325353176991869081, −25.16425050051067020524572529329, −23.94715928711648748027885324736, −22.587787272701135242651753555503, −21.24842764989950749738319601070, −20.52394480422918311745725924579, −19.99560424188993987258040532826, −18.92804667835009277609402497210, −17.94256486847020382815185860671, −16.4522154241824306430468804651, −15.93963700943372054104474679130, −14.97036637292408104373388433457, −13.294950562971487374366951826981, −12.39867210930412274804004269393, −11.08297196591747815410940991959, −10.10413314787119519506926549442, −9.12011193616051733817171872486, −8.06832301883160349421756623358, −7.53664336478487479932103530296, −5.23150483257360919231913712724, −3.80003532277246323727580207800, −2.8344382436407718395799091737, −1.12037558266479928664457215084, 0.727161141466245279126847537926, 2.34492557055093548040054187660, 3.585379996347507884057673118120, 5.748233775732020705307168187702, 7.02368194736489823724241192305, 7.736659141974580116866193330819, 8.61243333827577975100353640973, 9.82951463091063692288845392286, 11.04148366591940579603056063527, 12.07987206982349110769367974392, 13.73833433757921084843256134828, 14.512090086582221330008099445970, 15.588992367557744485959314569263, 16.35238633928191629856092362220, 18.00562024409698651234873415842, 18.617057353806559604506138501222, 19.24548412281166451364544936368, 20.260485300373396849102850770525, 21.26364314375638666936482833756, 23.176434914874309278489206719881, 23.69072892084011106037511683794, 24.71449240879506141163480912525, 25.75014375183640131744268251766, 26.43561678327639540568941814180, 27.05080492451488489429658353697

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