Properties

Label 1-181-181.3-r0-0-0
Degree $1$
Conductor $181$
Sign $0.332 + 0.943i$
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  + (−0.374 + 0.927i)2-s + (−0.882 + 0.469i)3-s + (−0.719 − 0.694i)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 + (0.559 − 0.829i)9-s + (0.559 − 0.829i)10-s + (−0.241 + 0.970i)11-s + (0.961 + 0.275i)12-s + (0.990 + 0.139i)13-s + (0.990 − 0.139i)14-s + (0.961 − 0.275i)15-s + (0.0348 + 0.999i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (−0.374 + 0.927i)2-s + (−0.882 + 0.469i)3-s + (−0.719 − 0.694i)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 + (0.559 − 0.829i)9-s + (0.559 − 0.829i)10-s + (−0.241 + 0.970i)11-s + (0.961 + 0.275i)12-s + (0.990 + 0.139i)13-s + (0.990 − 0.139i)14-s + (0.961 − 0.275i)15-s + (0.0348 + 0.999i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.332 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $0.332 + 0.943i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ 0.332 + 0.943i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3937764745 + 0.2787565754i\)
\(L(\frac12)\) \(\approx\) \(0.3937764745 + 0.2787565754i\)
\(L(1)\) \(\approx\) \(0.4852883674 + 0.2316695833i\)
\(L(1)\) \(\approx\) \(0.4852883674 + 0.2316695833i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (-0.374 + 0.927i)T \)
3 \( 1 + (-0.882 + 0.469i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.241 + 0.970i)T \)
13 \( 1 + (0.990 + 0.139i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.997 + 0.0697i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.848 + 0.529i)T \)
41 \( 1 + (0.438 - 0.898i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (0.961 + 0.275i)T \)
53 \( 1 + (0.438 + 0.898i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.766 - 0.642i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (0.669 + 0.743i)T \)
73 \( 1 + (0.766 - 0.642i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (-0.615 - 0.788i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−27.27934210120512732113118695446, −26.58979282133502360960169593745, −25.23613850268106789333113154320, −24.02216277724322740021435081194, −23.06171288832914470131594098200, −22.26210229160131538639773334691, −21.564678365518537735239471584326, −20.09799347120868507029546037209, −19.235462644269349509410271032924, −18.39945317344044535775147373388, −17.91569813146175522715175608286, −16.15142468449606913110839399258, −15.95690434104272229980497720375, −13.8607823169794064598519968318, −12.86182388872931616913939951476, −11.88589599382626926365336857116, −11.32430724162693520399354476296, −10.37934421031719125482257924191, −8.8384487053372969202883130473, −7.96377437339102337445427226608, −6.58024787372371966223445035102, −5.25889962754455513708651920160, −3.79829573418917103422635213614, −2.59090775778483162115889385519, −0.77789411183561563454030531235, 0.86191323062896231657293744654, 3.98202967864857834725143675213, 4.526803070623355101187060685248, 6.01480135424533830290483876907, 6.98323171400399317800555703432, 7.949314707589611155098286180586, 9.36605389065252452246121675930, 10.31218417957580626697998341321, 11.31000926708031724308693954267, 12.62808942151356402872759711908, 13.77220563459567285435777403577, 15.31909482797438525997182087666, 15.806740939805430504983185141326, 16.59122247339285918171809260909, 17.580026045442943964705937083165, 18.420604419880499876451374514323, 19.73523988844293291221149069099, 20.55292407714551832288415888315, 22.28547210187605246998188918963, 22.96188375737799431103164174318, 23.58142539885932468685573372635, 24.357801781586787541215633088407, 25.91914812707916096459562346411, 26.528520012846876153596672168885, 27.39298703106009095966439382575

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