Properties

Label 1-161-161.123-r0-0-0
Degree $1$
Conductor $161$
Sign $0.163 - 0.986i$
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  + (0.723 − 0.690i)2-s + (0.981 + 0.189i)3-s + (0.0475 − 0.998i)4-s + (−0.786 − 0.618i)5-s + (0.841 − 0.540i)6-s + (−0.654 − 0.755i)8-s + (0.928 + 0.371i)9-s + (−0.995 + 0.0950i)10-s + (0.723 + 0.690i)11-s + (0.235 − 0.971i)12-s + (0.415 − 0.909i)13-s + (−0.654 − 0.755i)15-s + (−0.995 − 0.0950i)16-s + (−0.888 − 0.458i)17-s + (0.928 − 0.371i)18-s + (−0.888 + 0.458i)19-s + ⋯
L(s)  = 1  + (0.723 − 0.690i)2-s + (0.981 + 0.189i)3-s + (0.0475 − 0.998i)4-s + (−0.786 − 0.618i)5-s + (0.841 − 0.540i)6-s + (−0.654 − 0.755i)8-s + (0.928 + 0.371i)9-s + (−0.995 + 0.0950i)10-s + (0.723 + 0.690i)11-s + (0.235 − 0.971i)12-s + (0.415 − 0.909i)13-s + (−0.654 − 0.755i)15-s + (−0.995 − 0.0950i)16-s + (−0.888 − 0.458i)17-s + (0.928 − 0.371i)18-s + (−0.888 + 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.163 - 0.986i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (123, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.163 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.479339500 - 1.254902083i\)
\(L(\frac12)\) \(\approx\) \(1.479339500 - 1.254902083i\)
\(L(1)\) \(\approx\) \(1.543981100 - 0.8149224790i\)
\(L(1)\) \(\approx\) \(1.543981100 - 0.8149224790i\)

Euler product

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

−27.57889330019512274222755169005, −26.60467379895420044697291207243, −26.08916281654076313803342099035, −25.05369132862603151074680754831, −24.02662289069630897029821777839, −23.49110076504009177304155068088, −22.07879510885051678145442116637, −21.46488026633212681258523986518, −20.12268933947809685304652561006, −19.25658307982080067398828665129, −18.24893244717971517300117368165, −16.80084541332048727601899754685, −15.70381821830842682131148582376, −14.934517210192066821319263475056, −14.113357373017237791544998162167, −13.24550139329205201821569571125, −12.010667610313679855963234975702, −10.956368519346482084766575050929, −8.97557984740004159028250033434, −8.3057559592912536121606554063, −7.0330327479730732559492899942, −6.38134027972608204772596927397, −4.29523538086490638549192589101, −3.663324006987872664927823382977, −2.34535956728986987803590784107, 1.449869031769044864987917090051, 2.91831587591274935752450077826, 4.06852914351450076777647508402, 4.79675835046749318535205153909, 6.61007832253307487819896433628, 8.089751003388640816940371778698, 9.12533536559116194997424620948, 10.21563065242465191955687422991, 11.46686755739779266101087193283, 12.61689089164375007825380509250, 13.28474048994729001982113316955, 14.58899470246953439801811724918, 15.27247538213118264738992684255, 16.195056871842737558943942021012, 17.97840532877286362598008193469, 19.288366477920200889952423021667, 19.9431107456539031049705313362, 20.5199711163607006054285127134, 21.51998552902754128104295216709, 22.69853501813124090997179910506, 23.5379878819800610403081036379, 24.745114083040944898609141118819, 25.2609992426561747633344646666, 26.965273414323268997382445971497, 27.579606907287184967668346954961

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