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Properties

Label 1-181-181.15-r0-0-0
Degree $1$
Conductor $181$
Sign $0.931 + 0.364i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $0.931 + 0.364i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ 0.931 + 0.364i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.743468889 + 0.3293730306i\)
\(L(\frac12)\) \(\approx\) \(1.743468889 + 0.3293730306i\)
\(L(1)\) \(\approx\) \(1.544392094 + 0.3165229763i\)
\(L(1)\) \(\approx\) \(1.544392094 + 0.3165229763i\)

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

−27.40269697736852455108037357977, −26.38660892013969219351063988040, −25.36669013440983802469924706362, −24.52160995921544706962902964854, −22.95558173029306761998480818610, −22.230956121956090825703655171732, −21.6199865395517180113079354715, −20.36496430111582053162259890491, −19.71049339533979854743294793805, −18.6133699808624040015782570415, −18.26059653522022803362928372089, −16.09493219915871211774503786452, −15.07180548644739707310080821705, −14.38385047276391200616113392228, −13.5102029091016345733407795668, −12.29992754567583651522713152844, −11.29081889848376885969631120354, −10.0460773316567075482911208297, −9.39882596236734705924565296173, −8.16734633602562419527200244271, −6.57660620630165009493136855602, −5.28530666681383924856771721399, −3.47163109415724122641793593553, −3.229853444378677997964440501037, −1.83711397805137401672094848158, 1.43325519121453594065830172118, 3.71752953983274921647553135820, 4.04231809554654458269450778447, 5.77149988910322619618251413311, 7.03812918549436792378758486846, 7.86334355598069031260579329314, 9.01227299357757244003405512556, 9.65228870621324422957112647375, 11.889074669085281701399326349851, 12.82647977604676974851416177895, 13.73568668318512110427871306035, 14.32437053904846742240004043832, 15.645633419816198370898748144625, 16.48891414544482849536402232297, 17.26752154216218474370805505591, 18.666744423962204177737352505988, 19.78733548088022472871691676215, 20.58322576245134860698901962095, 21.569854196648779903044250522066, 22.83851667178742910160866341811, 23.95496261664011068151633097559, 24.32274179923427654567655145938, 25.51944636551001462631968314812, 26.03203114100583072948530673984, 27.053583699278151805690013597598

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