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Properties

Label 1-161-161.129-r0-0-0
Degree $1$
Conductor $161$
Sign $0.679 + 0.733i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $0.679 + 0.733i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (129, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ 0.679 + 0.733i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7501139508 + 0.3275112300i\)
\(L(\frac12)\) \(\approx\) \(0.7501139508 + 0.3275112300i\)
\(L(1)\) \(\approx\) \(0.7593832791 + 0.3105293715i\)
\(L(1)\) \(\approx\) \(0.7593832791 + 0.3105293715i\)

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

−28.00073403434483916564958297442, −26.809676661996878383948537655989, −26.12354104800628985127670359934, −24.45464815346168821920606414856, −23.40093298964342588740355726432, −22.74595336644431833402991969726, −21.624494288039427969677162400526, −21.30568864834988203001787803392, −19.4150565956843067694693310576, −18.88738782243835042981958183412, −17.994959769751853919900529718617, −17.14638213487411671341907730764, −15.7980586646066796231038522598, −14.152483330035955822189913189091, −13.510653875696744781440353116573, −12.24905994124346692927525799582, −11.20523038653265027444307536964, −10.75725663920101407878671155610, −9.54497177369639339870445724832, −8.029359383582873487054512307430, −6.500608849401566996273067424496, −5.583004212880238386459385790367, −4.05727780119073368909613003288, −2.71403069595140913738688779304, −1.224769848098607249462486986573, 1.00781207636517584853344858608, 3.86138722182280614784619964011, 5.04930199526407473375126229564, 5.623712144905059333335253008841, 6.93832627806639514435662848268, 8.17164677492411049162928894312, 9.46628636770440731895448340479, 10.22427380943501919436018036875, 12.031741950504899950327974033998, 12.73823847530162116193312322118, 13.92640684121421451751233543685, 15.345910509215485527632252135604, 16.04481941756359835312934397370, 16.90640394584955878435362093816, 17.72194868399979924917670182861, 18.51212474665313053630936836045, 20.37493670158493395812880681178, 21.17486891683384037694991378764, 22.51800326479472368463451853140, 23.04366199708674401909865992751, 24.01818034419632437178461353152, 24.975277751138279860179721281486, 25.75901582403771380192483072969, 27.1664612837843836630897623375, 27.78887797538822320120083412100

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