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Properties

Label 1-161-161.158-r1-0-0
Degree $1$
Conductor $161$
Sign $0.989 - 0.144i$
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  + (0.235 − 0.971i)2-s + (−0.327 + 0.945i)3-s + (−0.888 − 0.458i)4-s + (−0.928 − 0.371i)5-s + (0.841 + 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (−0.580 + 0.814i)10-s + (−0.235 − 0.971i)11-s + (0.723 − 0.690i)12-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.580 + 0.814i)16-s + (−0.0475 + 0.998i)17-s + (−0.786 + 0.618i)18-s + (−0.0475 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.235 − 0.971i)2-s + (−0.327 + 0.945i)3-s + (−0.888 − 0.458i)4-s + (−0.928 − 0.371i)5-s + (0.841 + 0.540i)6-s + (−0.654 + 0.755i)8-s + (−0.786 − 0.618i)9-s + (−0.580 + 0.814i)10-s + (−0.235 − 0.971i)11-s + (0.723 − 0.690i)12-s + (0.415 + 0.909i)13-s + (0.654 − 0.755i)15-s + (0.580 + 0.814i)16-s + (−0.0475 + 0.998i)17-s + (−0.786 + 0.618i)18-s + (−0.0475 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(17.3018\)
Root analytic conductor: \(17.3018\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (1:\ ),\ 0.989 - 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.072965539 - 0.07785613151i\)
\(L(\frac12)\) \(\approx\) \(1.072965539 - 0.07785613151i\)
\(L(1)\) \(\approx\) \(0.8026638689 - 0.1898705674i\)
\(L(1)\) \(\approx\) \(0.8026638689 - 0.1898705674i\)

Euler product

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

−27.4759600190161784230307626466, −26.470859867387289119332313047259, −25.26521597553839499544052829806, −24.795656477218822867169319403468, −23.44907202036847654491614992305, −23.05670986943735689923924200707, −22.401369650287264216684415939704, −20.651160471394322264971111160148, −19.43727773370586438227107168058, −18.32883560627182391088919610819, −17.83444530875199716351033010369, −16.55911190667866726006922065537, −15.58351805489832582420621611669, −14.64325752834126806912452671586, −13.51960863717930563113436918275, −12.500515881992307983381708941801, −11.70581957676196453279773601558, −10.148465465825356572143531137052, −8.38342412055398204539613815900, −7.653006881531639277085579802314, −6.83793431309414943811739544421, −5.66857448115444502355587454363, −4.39107652356811729152070526471, −2.85570580299251660667075893639, −0.583631613105927302502095029409, 0.85178989235038226638564682079, 2.99620804245687900438297904270, 4.04237421223257880841988294262, 4.84413642649619079985747687627, 6.20061248061647774123678152717, 8.39912042370052072395255705932, 9.06334539432445679789657688351, 10.46748252028849378362959652884, 11.25871094342131867350937657302, 11.979050100630506644025042706032, 13.26958697840125239578964972812, 14.47949454832895331689502701289, 15.56959193139077210583474481803, 16.480373498335780221653611092160, 17.66068009560998524608090211404, 19.07562491539949513665211930000, 19.71119676900214990101920929264, 20.88268005028868975274871993006, 21.51566877322746727045048894411, 22.4277053757109043283789372176, 23.583814970093522910623456297911, 23.993519219878035007565507381114, 26.140172917481094609934882656438, 26.82506954628049174943883149953, 27.70903735097658133784552692900

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