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Properties

Label 1-161-161.159-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.328 - 0.944i$
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  + (0.580 − 0.814i)2-s + (−0.235 + 0.971i)3-s + (−0.327 − 0.945i)4-s + (0.0475 − 0.998i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (−0.888 − 0.458i)9-s + (−0.786 − 0.618i)10-s + (−0.580 − 0.814i)11-s + (0.995 − 0.0950i)12-s + (0.142 − 0.989i)13-s + (0.959 + 0.281i)15-s + (−0.786 + 0.618i)16-s + (0.981 − 0.189i)17-s + (−0.888 + 0.458i)18-s + (0.981 + 0.189i)19-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (−0.235 + 0.971i)3-s + (−0.327 − 0.945i)4-s + (0.0475 − 0.998i)5-s + (0.654 + 0.755i)6-s + (−0.959 − 0.281i)8-s + (−0.888 − 0.458i)9-s + (−0.786 − 0.618i)10-s + (−0.580 − 0.814i)11-s + (0.995 − 0.0950i)12-s + (0.142 − 0.989i)13-s + (0.959 + 0.281i)15-s + (−0.786 + 0.618i)16-s + (0.981 − 0.189i)17-s + (−0.888 + 0.458i)18-s + (0.981 + 0.189i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.328 - 0.944i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.328 - 0.944i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6791432927 - 0.9556377870i\)
\(L(\frac12)\) \(\approx\) \(0.6791432927 - 0.9556377870i\)
\(L(1)\) \(\approx\) \(0.9891223480 - 0.5919066560i\)
\(L(1)\) \(\approx\) \(0.9891223480 - 0.5919066560i\)

Euler product

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

−28.15542881181074849067080613590, −26.68410245820141730065683795862, −25.85009533223431402789312858154, −25.24632748178025475815544169966, −23.96732197224296328456914835078, −23.391900963027422809020114697769, −22.5460312589986014720942982687, −21.6535670515376099213870863885, −20.29930720140951760255213636321, −18.73297504978783752474694143288, −18.25211092786372398143596301253, −17.21103300439827963068267472705, −16.14942851715023907095334373720, −14.82234478204813415639380242251, −14.14038272542038963463249717427, −13.14314907898708791847997475849, −12.11469950366008844233826886827, −11.1394969478921406586983801403, −9.464143439800395515431281621832, −7.80792259868353861068362550474, −7.23396445657916744106234076481, −6.25761385090414176290512128942, −5.18131134674337522191025141284, −3.48734028252178458554422133436, −2.174599840525354944097136792733, 0.872044548076312093775085134696, 2.94580473696608844768534806048, 3.96370639394252423346095186618, 5.36712911037103751705432113916, 5.62197373752251264838810575947, 8.15097322146123558008528178926, 9.338443120778117467604616331353, 10.189715265168911924807776398473, 11.23919620307012582815349189348, 12.22930456666731472783546643639, 13.27682514260339124204601355113, 14.33512591635292025465396373777, 15.58185379536562512300793932495, 16.30879907026587921292106672188, 17.57113311207986974540721957911, 18.83611510236391747941667186611, 20.20171697621391254275532988949, 20.66263441530312639733019711634, 21.50094361480108909335608355791, 22.4451729383081879452152954426, 23.38845790106485771006985008155, 24.303265119227857199870987845109, 25.52528275206042835660977679805, 27.096021880284268368368875433757, 27.531044595983651003835295300525

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