Properties

Label 1-185-185.157-r1-0-0
Degree $1$
Conductor $185$
Sign $0.0656 - 0.997i$
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  + (−0.342 − 0.939i)2-s + (−0.342 + 0.939i)3-s + (−0.766 + 0.642i)4-s + 6-s + (−0.984 + 0.173i)7-s + (0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.342 − 0.939i)12-s + (−0.642 − 0.766i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)18-s + (0.939 + 0.342i)19-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.342 + 0.939i)3-s + (−0.766 + 0.642i)4-s + 6-s + (−0.984 + 0.173i)7-s + (0.866 + 0.5i)8-s + (−0.766 − 0.642i)9-s + (−0.5 + 0.866i)11-s + (−0.342 − 0.939i)12-s + (−0.642 − 0.766i)13-s + (0.5 + 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.642 + 0.766i)17-s + (−0.342 + 0.939i)18-s + (0.939 + 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0656 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0656 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.0656 - 0.997i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ 0.0656 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3655635265 - 0.3423063244i\)
\(L(\frac12)\) \(\approx\) \(0.3655635265 - 0.3423063244i\)
\(L(1)\) \(\approx\) \(0.5652106840 - 0.06838920995i\)
\(L(1)\) \(\approx\) \(0.5652106840 - 0.06838920995i\)

Euler product

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

−26.79541731500880760534868805265, −26.27958352725640188523914040968, −25.14820186712604386411707880558, −24.345987908669246048857546703258, −23.69313052904821425953051944621, −22.64941492979194746861460224798, −21.973449552624761522751209469939, −19.98632130364945006354255827195, −19.192044255472878202796970499988, −18.405156580040115274793572289953, −17.525331353702682644962760909093, −16.35964913176314951588012957006, −15.94837005481353717221596790012, −14.19183412816782054698844318308, −13.62106376036548200110742586362, −12.58339896531375181273378486739, −11.26447431624175354371493481259, −9.9275712074734839082558677042, −8.83272312737607762066102244140, −7.65074481734942038901278100442, −6.76144924709311447745651732820, −5.97180833422641625126315135290, −4.71791782073535337770297908773, −2.76139670824187433373649223336, −0.846831757664400100099332053280, 0.289644486794239014527000452145, 2.42161335594489161417518073019, 3.53498221237452869088509240368, 4.63963924064630964870208211168, 5.87523453657891699152228907715, 7.632453373112387137938543314042, 8.98267463572586789302247005716, 10.03751792829055134883105400971, 10.32016432442389649825690739775, 11.82287443968856627394417116129, 12.52518721429096239079232002707, 13.71204913026103686903538303515, 15.216041905878238865698571683267, 16.01964620286810336766868348528, 17.27975890612885302933022772523, 17.878742672616036936109286052259, 19.284518451055424427341626098528, 20.10685652326599439821256340405, 20.8810448890023114143299508723, 22.096265304583048702465095760254, 22.44951031614971732862059154761, 23.423699967469816740371387847366, 25.22903540440476438145564856891, 26.21317764138492828903266101969, 26.789771735456272895660288912221

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