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Properties

Label 1-161-161.2-r0-0-0
Degree $1$
Conductor $161$
Sign $-0.942 - 0.335i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(161\)    =    \(7 \cdot 23\)
Sign: $-0.942 - 0.335i$
Analytic conductor: \(0.747680\)
Root analytic conductor: \(0.747680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{161} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 161,\ (0:\ ),\ -0.942 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2547939007 - 1.476092347i\)
\(L(\frac12)\) \(\approx\) \(0.2547939007 - 1.476092347i\)
\(L(1)\) \(\approx\) \(0.8242029381 - 1.124449165i\)
\(L(1)\) \(\approx\) \(0.8242029381 - 1.124449165i\)

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

−27.702375732521100491799590578521, −27.03919991762852197727448518725, −26.20955170083247903479399966802, −25.45059883106962776578384753137, −24.47238504934021797295303763714, −23.06549527081076695567103951494, −22.39183964864706880398307767098, −21.728076142796921337971000108629, −20.74586039650633383339897169374, −19.43678161972857062010677756466, −18.10836122228846146895998843071, −17.046168618857268002115224135320, −16.065283576491398427245247896211, −15.17807592758742253806837795673, −14.370772284183005971735602615193, −13.676381051193452725918303977993, −12.03730520219949826617357562196, −10.898903849632233491087471035648, −9.75335564622137788548910930375, −8.491097749262001747867130374562, −7.414209979401727747906192112936, −6.04471969541751483166286563905, −5.146990298536048829872782246553, −3.53975051950768730767864255765, −3.06700790673175923845275963258, 1.15206194149950714490080266903, 2.12438240202309609843289364697, 3.73564857507615574083912600299, 5.01224759414143217760785388317, 6.19787341262629046983710140270, 7.58688202092690333361090110810, 9.034116711531247624687564150832, 9.792283683681338623533080234162, 11.808230788781643003836968891667, 11.99041036375337345439970507696, 13.1667865023027656782153767539, 13.95310697866289931220358243889, 14.93611407818001367291132893238, 16.543427697041007544252591863785, 17.67833800689239699655085144006, 18.78571068133183851063230597606, 19.63952549723965091165946779114, 20.45202810745423047939089954231, 21.226884200810187855738318491653, 22.66642300725795356785880139568, 23.4124887303874379153533184683, 24.44535013777912583313859587545, 24.92742806765278340826970978374, 26.36897763264677343944304716182, 27.91154243003390803244583903384

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