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Properties

Label 1-181-181.37-r0-0-0
Degree $1$
Conductor $181$
Sign $-0.590 + 0.806i$
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  + (0.615 + 0.788i)2-s + (0.848 + 0.529i)3-s + (−0.241 + 0.970i)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 + (0.438 + 0.898i)9-s + (−0.438 − 0.898i)10-s + (0.961 − 0.275i)11-s + (−0.719 + 0.694i)12-s + (−0.374 − 0.927i)13-s + (−0.374 + 0.927i)14-s + (−0.719 − 0.694i)15-s + (−0.882 − 0.469i)16-s + (−0.173 + 0.984i)17-s + ⋯
L(s)  = 1  + (0.615 + 0.788i)2-s + (0.848 + 0.529i)3-s + (−0.241 + 0.970i)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 + (0.438 + 0.898i)9-s + (−0.438 − 0.898i)10-s + (0.961 − 0.275i)11-s + (−0.719 + 0.694i)12-s + (−0.374 − 0.927i)13-s + (−0.374 + 0.927i)14-s + (−0.719 − 0.694i)15-s + (−0.882 − 0.469i)16-s + (−0.173 + 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 181 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(181\)
Sign: $-0.590 + 0.806i$
Analytic conductor: \(0.840560\)
Root analytic conductor: \(0.840560\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{181} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 181,\ (0:\ ),\ -0.590 + 0.806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7827305310 + 1.543202892i\)
\(L(\frac12)\) \(\approx\) \(0.7827305310 + 1.543202892i\)
\(L(1)\) \(\approx\) \(1.156150523 + 1.045669922i\)
\(L(1)\) \(\approx\) \(1.156150523 + 1.045669922i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad181 \( 1 \)
good2 \( 1 + (0.615 + 0.788i)T \)
3 \( 1 + (0.848 + 0.529i)T \)
5 \( 1 + (-0.978 - 0.207i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.961 - 0.275i)T \)
13 \( 1 + (-0.374 - 0.927i)T \)
17 \( 1 + (-0.173 + 0.984i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.559 - 0.829i)T \)
29 \( 1 + (0.913 - 0.406i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (0.0348 - 0.999i)T \)
41 \( 1 + (0.997 - 0.0697i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.719 - 0.694i)T \)
53 \( 1 + (0.997 + 0.0697i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.939 + 0.342i)T \)
67 \( 1 + (-0.104 - 0.994i)T \)
71 \( 1 + (-0.669 - 0.743i)T \)
73 \( 1 + (-0.939 - 0.342i)T \)
79 \( 1 + (-0.997 - 0.0697i)T \)
83 \( 1 + (-0.990 + 0.139i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (-0.961 - 0.275i)T \)
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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

−27.19653027171926530160814229834, −26.143978323967065468212253430050, −24.746514571570284332242869569725, −23.854212896715320472841284955549, −23.33843357523692775873313455396, −22.153047389407300540405834056612, −20.943264432671494116192911843486, −20.08462770090453185310707206193, −19.50870801205991324254103861558, −18.73100307036084168598871678494, −17.47673346132838644762550514554, −15.85392613187419634526139923750, −14.564746006533351576660529198764, −14.27410088317602352349685522948, −13.10460920196294067860076697119, −11.93342328469412214184859091138, −11.33828212223068950179279204727, −9.88862125770522528463016584335, −8.79956291178355904273223666380, −7.44007593124883077359713499080, −6.5903569426297404874439973350, −4.41065760831178507723558612898, −3.93374928458942762347725700440, −2.5222891683086269033863143876, −1.18109220883883820222275683562, 2.500067675268813697475341580290, 3.807291405992214688711943277390, 4.54342147285906235254781451445, 5.86773808825207577968788165945, 7.38082535810299602484368106232, 8.52076413808698772731634960024, 8.743780095075316647720909559696, 10.65768716869955558723712614943, 12.06371870538932239481248037444, 12.79152575582405771747279690342, 14.304868913839328257732862983952, 14.90530236798715153928657348828, 15.58824205048719277673092859657, 16.52480173669764251061761925060, 17.70277918692957056075122772367, 19.14893974583347698627369887982, 19.96877158048267215218862999785, 21.15645949084873237923618150507, 21.900209937740934094809483274141, 22.81034301665122232972250668917, 24.09026765518885906372378001325, 24.76069175572556842189361236101, 25.50439610239462383581962517351, 26.6899609014457445734698966139, 27.37536698684875770153257153387

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