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Properties

Label 1-185-185.182-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.902 - 0.430i$
Analytic cond. $19.8810$
Root an. cond. $19.8810$
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  + (−0.984 + 0.173i)2-s + (−0.984 − 0.173i)3-s + (0.939 − 0.342i)4-s + 6-s + (−0.642 + 0.766i)7-s + (−0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.984 + 0.173i)12-s + (0.342 + 0.939i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (−0.984 − 0.173i)18-s + (−0.173 + 0.984i)19-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.984 − 0.173i)3-s + (0.939 − 0.342i)4-s + 6-s + (−0.642 + 0.766i)7-s + (−0.866 + 0.5i)8-s + (0.939 + 0.342i)9-s + (−0.5 − 0.866i)11-s + (−0.984 + 0.173i)12-s + (0.342 + 0.939i)13-s + (0.5 − 0.866i)14-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + (−0.984 − 0.173i)18-s + (−0.173 + 0.984i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.902 - 0.430i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (182, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.902 - 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001735743167 + 0.007670715222i\)
\(L(\frac12)\) \(\approx\) \(0.001735743167 + 0.007670715222i\)
\(L(1)\) \(\approx\) \(0.4231703395 + 0.04672981749i\)
\(L(1)\) \(\approx\) \(0.4231703395 + 0.04672981749i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.984 + 0.173i)T \)
3 \( 1 + (-0.984 - 0.173i)T \)
7 \( 1 + (-0.642 + 0.766i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (0.342 + 0.939i)T \)
17 \( 1 + (0.342 - 0.939i)T \)
19 \( 1 + (-0.173 + 0.984i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + (-0.642 - 0.766i)T \)
59 \( 1 + (-0.766 + 0.642i)T \)
61 \( 1 + (-0.939 + 0.342i)T \)
67 \( 1 + (-0.642 + 0.766i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.766 - 0.642i)T \)
83 \( 1 + (-0.342 + 0.939i)T \)
89 \( 1 + (-0.766 + 0.642i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−27.53112003744494276694065221289, −26.50656933915120544332841977861, −25.80620797096109427439005472254, −24.63498956133545339527740176185, −23.43193916642980492743430490668, −22.76384243423272594679475644576, −21.47854085417085280546173233560, −20.57321808465115008209326075283, −19.58286798222501585831496069951, −18.5444003486117098375805532742, −17.4299978971642583275127799765, −17.09756208954484664295605152008, −15.83643984977893718603020715682, −15.235247520446118979691118388905, −13.09522996295900357375958658268, −12.43133015144694735305394983739, −11.0758917794480458183866645427, −10.36202399137904207507718075906, −9.65696087439285525925721307860, −8.07965591588111001653692849371, −6.9812885013756711272189025153, −6.12147837255287774320675893549, −4.54233310094947742157468321364, −3.01354609047521244024142600795, −1.16893889787005936712124050699, 0.00509826044300079497125422285, 1.46644554687432738741412364212, 3.068168924416104674900640093568, 5.2126442777857824924902691128, 6.15696572249820371455330601659, 7.01293117955296083432598351819, 8.36886581506738149638597921681, 9.467747871213493831284333866842, 10.48691268489626808807302954736, 11.53844695684383995662427864255, 12.23899757331251985335092921627, 13.66151807930597948417255759765, 15.275485397170995451131216019698, 16.28598587030725011334882376836, 16.61287072266506706151060755748, 17.99809083521955467688889607508, 18.736980950462664414359235963418, 19.23722955895563213316113081494, 20.92359700702082316561654813764, 21.63680128837789904449938188072, 22.94648986904706244986550339910, 23.804265180960364446115027035460, 24.796453115330976053871196437062, 25.58060556921238568687776410370, 26.810498468125416415181308210978

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