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Properties

Label 1-181-181.12-r0-0-0
Degree $1$
Conductor $181$
Sign $0.152 + 0.988i$
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  + (−0.438 + 0.898i)2-s + (0.961 + 0.275i)3-s + (−0.615 − 0.788i)4-s + (−0.104 + 0.994i)5-s + (−0.669 + 0.743i)6-s + (0.5 − 0.866i)7-s + (0.978 − 0.207i)8-s + (0.848 + 0.529i)9-s + (−0.848 − 0.529i)10-s + (0.990 − 0.139i)11-s + (−0.374 − 0.927i)12-s + (0.559 − 0.829i)13-s + (0.559 + 0.829i)14-s + (−0.374 + 0.927i)15-s + (−0.241 + 0.970i)16-s + (−0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.438 + 0.898i)2-s + (0.961 + 0.275i)3-s + (−0.615 − 0.788i)4-s + (−0.104 + 0.994i)5-s + (−0.669 + 0.743i)6-s + (0.5 − 0.866i)7-s + (0.978 − 0.207i)8-s + (0.848 + 0.529i)9-s + (−0.848 − 0.529i)10-s + (0.990 − 0.139i)11-s + (−0.374 − 0.927i)12-s + (0.559 − 0.829i)13-s + (0.559 + 0.829i)14-s + (−0.374 + 0.927i)15-s + (−0.241 + 0.970i)16-s + (−0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.152 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $0.152 + 0.988i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ 0.152 + 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9788370371 + 0.8390718106i\)
\(L(\frac12)\) \(\approx\) \(0.9788370371 + 0.8390718106i\)
\(L(1)\) \(\approx\) \(1.015704672 + 0.5790540740i\)
\(L(1)\) \(\approx\) \(1.015704672 + 0.5790540740i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (-0.438 + 0.898i)T \)
3 \( 1 + (0.961 + 0.275i)T \)
5 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (0.990 - 0.139i)T \)
13 \( 1 + (0.559 - 0.829i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 - T \)
23 \( 1 + (0.882 + 0.469i)T \)
29 \( 1 + (-0.978 + 0.207i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.719 + 0.694i)T \)
41 \( 1 + (-0.0348 - 0.999i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.374 + 0.927i)T \)
53 \( 1 + (-0.0348 + 0.999i)T \)
59 \( 1 + (-0.809 - 0.587i)T \)
61 \( 1 + (-0.173 + 0.984i)T \)
67 \( 1 + (0.669 - 0.743i)T \)
71 \( 1 + (-0.913 - 0.406i)T \)
73 \( 1 + (0.173 - 0.984i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.997 - 0.0697i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.990 - 0.139i)T \)
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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

−27.265477106247396550583599347034, −26.27255406801783632450298919877, −25.06444239839317200165269620990, −24.64712974003513290153659713058, −23.27112822399560860319315775020, −21.77396340850060613738393762425, −21.06261631597820178523005190708, −20.33333804755415715457480131774, −19.39855743150912385923765552305, −18.68121283378598437318901128241, −17.592716053062461144474018380616, −16.51268256364665866401430486216, −15.20480631394212559526623070478, −13.99778477462416519560387694164, −13.03167951815465387534311778734, −12.14807040512654378688777071921, −11.28528905020454041831742601499, −9.543003978686849628993325112871, −8.811324475142757954854940617204, −8.40175958939194417132126722420, −6.82952585883023665982941098202, −4.77297091218157518645017210715, −3.84174678075154113477338998635, −2.27732235050629805160783785803, −1.40833128057333071937879776003, 1.63785293535018754991933919181, 3.50736559286307740557120129412, 4.46194059520922018982995568222, 6.26045006725914106407780966837, 7.22504311150353938393408009402, 8.128738077765939856157599895858, 9.1054503234737105813467967351, 10.381122932196865914446374684301, 10.96942875391888900024216597753, 13.263609702962210924363703657582, 13.98552006973636745055870121180, 14.98906555537728457074697544688, 15.34645460782797288166871356931, 16.872126941746215691307154117756, 17.66541646511507449350217398588, 18.898352976708434932871303046339, 19.53206658906449631396080701855, 20.56597920747062221538341630180, 21.938620231888648666519632577359, 22.855590681531157691827694173387, 23.90187340963300412413571975795, 24.89456814872931377523821718966, 25.78601710247466843034902315594, 26.41913714723029224937164779530, 27.29546436958453581553424017385

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