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Properties

Label 1-161-161.156-r0-0-0
Degree $1$
Conductor $161$
Sign $0.994 - 0.103i$
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  + (0.0475 − 0.998i)2-s + (0.928 + 0.371i)3-s + (−0.995 − 0.0950i)4-s + (0.235 + 0.971i)5-s + (0.415 − 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (0.981 − 0.189i)10-s + (0.0475 + 0.998i)11-s + (−0.888 − 0.458i)12-s + (−0.654 − 0.755i)13-s + (−0.142 + 0.989i)15-s + (0.981 + 0.189i)16-s + (0.580 + 0.814i)17-s + (0.723 − 0.690i)18-s + (0.580 − 0.814i)19-s + ⋯
L(s)  = 1  + (0.0475 − 0.998i)2-s + (0.928 + 0.371i)3-s + (−0.995 − 0.0950i)4-s + (0.235 + 0.971i)5-s + (0.415 − 0.909i)6-s + (−0.142 + 0.989i)8-s + (0.723 + 0.690i)9-s + (0.981 − 0.189i)10-s + (0.0475 + 0.998i)11-s + (−0.888 − 0.458i)12-s + (−0.654 − 0.755i)13-s + (−0.142 + 0.989i)15-s + (0.981 + 0.189i)16-s + (0.580 + 0.814i)17-s + (0.723 − 0.690i)18-s + (0.580 − 0.814i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.103i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.994 - 0.103i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.994 - 0.103i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.420704592 - 0.07391024566i\)
\(L(\frac12)\) \(\approx\) \(1.420704592 - 0.07391024566i\)
\(L(1)\) \(\approx\) \(1.302553949 - 0.1801652588i\)
\(L(1)\) \(\approx\) \(1.302553949 - 0.1801652588i\)

Euler product

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

−27.2807031609930798891231066171, −26.811967019961563301134866349307, −25.55899187654464649086632002939, −24.89817572534740889879538134767, −24.195600405919011069506341039405, −23.44537488783146400129639763811, −21.86736181680724451841139640913, −21.05185305512443021515457831791, −19.84238565616853846166759051148, −18.83130625031552026778276647365, −17.9169134712478149084306715146, −16.516841631406612021192586494455, −16.11262824257344580009793707357, −14.531823028738340582449460025086, −13.99806862701835611301179646956, −13.00118508870048992583218744339, −12.0354884081294599179581506677, −9.78137976815569543176694683340, −9.03286594346640201727372789201, −8.1430815082517377024115642875, −7.15135387357037258185853312302, −5.80556257859662917601097413231, −4.607661752882237424251530495462, −3.26431819063280846926017096754, −1.2652681494707855042086673621, 1.97250495825826845907523930074, 2.890320057649217105301472296830, 3.9473596307439596418802574851, 5.277923933599854466777688285, 7.20536465352944727058037893653, 8.32423817102303006301874192515, 9.90971714896788151072679802513, 9.99308615374883110367548964223, 11.38249297416473502513917916437, 12.721210415011739025264872798641, 13.65941401603721846166565064933, 14.741736917191200011491023715290, 15.23732069065949372653631874619, 17.2163248894178738577708432582, 18.18858180136528562306183904815, 19.17184125928332886402265984400, 19.96853569660641165789013677997, 20.80897781313581157080039083963, 21.9318456766519415837800073521, 22.40049228658890120492442536855, 23.67342643977527479405448076272, 25.22710057719370947878368088801, 26.028349097010121807567376736877, 26.87722088269712396053306522462, 27.68279723850672265413124663713

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