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Properties

Label 1-161-161.128-r0-0-0
Degree $1$
Conductor $161$
Sign $0.920 - 0.390i$
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  + (0.928 − 0.371i)2-s + (−0.995 − 0.0950i)3-s + (0.723 − 0.690i)4-s + (−0.327 + 0.945i)5-s + (−0.959 + 0.281i)6-s + (0.415 − 0.909i)8-s + (0.981 + 0.189i)9-s + (0.0475 + 0.998i)10-s + (0.928 + 0.371i)11-s + (−0.786 + 0.618i)12-s + (0.841 − 0.540i)13-s + (0.415 − 0.909i)15-s + (0.0475 − 0.998i)16-s + (0.235 − 0.971i)17-s + (0.981 − 0.189i)18-s + (0.235 + 0.971i)19-s + ⋯
L(s)  = 1  + (0.928 − 0.371i)2-s + (−0.995 − 0.0950i)3-s + (0.723 − 0.690i)4-s + (−0.327 + 0.945i)5-s + (−0.959 + 0.281i)6-s + (0.415 − 0.909i)8-s + (0.981 + 0.189i)9-s + (0.0475 + 0.998i)10-s + (0.928 + 0.371i)11-s + (−0.786 + 0.618i)12-s + (0.841 − 0.540i)13-s + (0.415 − 0.909i)15-s + (0.0475 − 0.998i)16-s + (0.235 − 0.971i)17-s + (0.981 − 0.189i)18-s + (0.235 + 0.971i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.920 - 0.390i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.920 - 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.445679390 - 0.2936913658i\)
\(L(\frac12)\) \(\approx\) \(1.445679390 - 0.2936913658i\)
\(L(1)\) \(\approx\) \(1.350529081 - 0.2183154538i\)
\(L(1)\) \(\approx\) \(1.350529081 - 0.2183154538i\)

Euler product

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

−28.14572085729211776387933599688, −26.93852023719700425675405579929, −25.69753785534502818442156889151, −24.41786813505948065298004787587, −23.978570848751675190107520048306, −23.11205246225282858682494083242, −22.07981970398751410513013094679, −21.326156140383827503045713806488, −20.325149333105633408312140901912, −19.077810604809165942085293688201, −17.42776643771223490150136295476, −16.761533169071566356713255053168, −16.00402175281960190397317033852, −15.03582296658008775982457012567, −13.55072586344435218710081123610, −12.73623485748013891811508894616, −11.689854441672068479372428932612, −11.11541004902760052621013956060, −9.28280106805975353026268645559, −7.96578418431574287723180956475, −6.56398983832051731751928559947, −5.7580272668451586865258704518, −4.54426358597432490974143824082, −3.777270911037888829630939339632, −1.44486781501378630698223868481, 1.44438566232075750448317401818, 3.206271991296565957674905376528, 4.29368682712487738911620305688, 5.66387607078406557200050977023, 6.546968345931414919308289293741, 7.49531098518550269631862091289, 9.78992549890037959882007525887, 10.75530910242390796124182016430, 11.58655036978168928603202570299, 12.32289876784576809835344112995, 13.604772229479711068033722901550, 14.64938786083813830242415610869, 15.6473550129369754130431647980, 16.593152135945063260910973956518, 18.080560188376683289146014549508, 18.79704120343295059639521849712, 20.00371895019373621666924696700, 21.135320599702841086754287348257, 22.25683358278860317461473638301, 22.79783376072997745482538651961, 23.35496034661781728516571842958, 24.61147338994684336139733548839, 25.48753448991737861708729357173, 27.15468067015893535260735065725, 27.81408743152763451563421343899

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