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Properties

Label 1-181-181.34-r0-0-0
Degree $1$
Conductor $181$
Sign $-0.484 - 0.874i$
Analytic cond. $0.840560$
Root an. cond. $0.840560$
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  + (0.990 − 0.139i)2-s + (0.0348 − 0.999i)3-s + (0.961 − 0.275i)4-s + (−0.978 − 0.207i)5-s + (−0.104 − 0.994i)6-s + (−0.5 − 0.866i)7-s + (0.913 − 0.406i)8-s + (−0.997 − 0.0697i)9-s + (−0.997 − 0.0697i)10-s + (−0.719 − 0.694i)11-s + (−0.241 − 0.970i)12-s + (−0.615 + 0.788i)13-s + (−0.615 − 0.788i)14-s + (−0.241 + 0.970i)15-s + (0.848 − 0.529i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (0.990 − 0.139i)2-s + (0.0348 − 0.999i)3-s + (0.961 − 0.275i)4-s + (−0.978 − 0.207i)5-s + (−0.104 − 0.994i)6-s + (−0.5 − 0.866i)7-s + (0.913 − 0.406i)8-s + (−0.997 − 0.0697i)9-s + (−0.997 − 0.0697i)10-s + (−0.719 − 0.694i)11-s + (−0.241 − 0.970i)12-s + (−0.615 + 0.788i)13-s + (−0.615 − 0.788i)14-s + (−0.241 + 0.970i)15-s + (0.848 − 0.529i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.484 - 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $-0.484 - 0.874i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ -0.484 - 0.874i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8051066263 - 1.366757396i\)
\(L(\frac12)\) \(\approx\) \(0.8051066263 - 1.366757396i\)
\(L(1)\) \(\approx\) \(1.212377751 - 0.8443847583i\)
\(L(1)\) \(\approx\) \(1.212377751 - 0.8443847583i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (0.990 - 0.139i)T \)
3 \( 1 + (0.0348 - 0.999i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.719 - 0.694i)T \)
13 \( 1 + (-0.615 + 0.788i)T \)
17 \( 1 + (0.766 + 0.642i)T \)
19 \( 1 + T \)
23 \( 1 + (0.438 - 0.898i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (-0.882 + 0.469i)T \)
41 \( 1 + (0.559 + 0.829i)T \)
43 \( 1 + (-0.939 - 0.342i)T \)
47 \( 1 + (-0.241 - 0.970i)T \)
53 \( 1 + (0.559 - 0.829i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.173 + 0.984i)T \)
79 \( 1 + (0.559 - 0.829i)T \)
83 \( 1 + (-0.374 + 0.927i)T \)
89 \( 1 + (0.173 - 0.984i)T \)
97 \( 1 + (-0.719 + 0.694i)T \)
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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

−27.63290126619856276263263955270, −26.60296628913943281581956585044, −25.58655530933414197393946942051, −24.8304273413935068027218195076, −23.35094818461062003753316166026, −22.80805563937061282464141782893, −22.04668877361061352507061265950, −21.04311869729931618228511682058, −20.14428615343443563424249074596, −19.31388635331173052983878344922, −17.7414801087494660269815484885, −16.216058659036251330258488441937, −15.68747361389031735441880641174, −15.091967366256690695658308520180, −14.07102781245360030860985686172, −12.497952829395527210955697108898, −11.91943711229665180095334337795, −10.739950359647584038578084422, −9.68290027036077276490119040540, −8.13280683993113801827725139982, −7.10968252917759047276216306912, −5.43983324947303805885725115669, −4.88729326703988240389570075453, −3.35266239279902334212988055717, −2.827927426747500258618016301538, 0.95483830610172267818400928664, 2.74185036827839301712939546787, 3.76398416366548570654693168222, 5.10656048420291124792357425718, 6.487396556561691153464468691115, 7.34530059089522941399087412292, 8.23665606772284314768010916555, 10.21285706768618241945801754223, 11.43103364368820959961390561764, 12.16127177460518606415714366925, 13.1278435629448063996009812896, 13.89798064755281436015564846950, 14.93138732658539069477375568094, 16.270697260122141602184556642531, 16.86009214947717157983990327713, 18.71289741844805132960450672853, 19.38892917977108101957559780005, 20.12922071375573032903683075601, 21.159136591128365365143275936721, 22.615915696843647404530402741035, 23.23180982098206755162522752070, 24.04581920171889882635367635716, 24.52118814450838253311577894486, 25.94876990034330406838988685304, 26.75366689322040391904340692830

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