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Properties

Label 1-185-185.167-r0-0-0
Degree $1$
Conductor $185$
Sign $-0.654 + 0.755i$
Analytic cond. $0.859136$
Root an. cond. $0.859136$
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  + (0.173 + 0.984i)2-s + (0.984 + 0.173i)3-s + (−0.939 + 0.342i)4-s + i·6-s + (−0.642 + 0.766i)7-s + (−0.5 − 0.866i)8-s + (0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 + 0.173i)12-s + (−0.939 + 0.342i)13-s + (−0.866 − 0.5i)14-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)18-s + (−0.984 − 0.173i)19-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)2-s + (0.984 + 0.173i)3-s + (−0.939 + 0.342i)4-s + i·6-s + (−0.642 + 0.766i)7-s + (−0.5 − 0.866i)8-s + (0.939 + 0.342i)9-s + (0.5 + 0.866i)11-s + (−0.984 + 0.173i)12-s + (−0.939 + 0.342i)13-s + (−0.866 − 0.5i)14-s + (0.766 − 0.642i)16-s + (0.939 + 0.342i)17-s + (−0.173 + 0.984i)18-s + (−0.984 − 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.654 + 0.755i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ -0.654 + 0.755i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5865973075 + 1.283984635i\)
\(L(\frac12)\) \(\approx\) \(0.5865973075 + 1.283984635i\)
\(L(1)\) \(\approx\) \(0.9614678317 + 0.8761304820i\)
\(L(1)\) \(\approx\) \(0.9614678317 + 0.8761304820i\)

Euler product

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

−26.960305934897839064639531801, −26.148864216367713938921769606060, −24.99004901627142617076972221577, −23.910452106645551528176716858564, −22.93374826287489826671715692852, −21.835833805325507596415814997344, −21.00518888264465264496600821902, −19.94459571492934179717024159680, −19.446243398049632752844541471305, −18.64600925771248265268071698083, −17.32668016227599844013873705463, −16.10798960774972931880279648894, −14.43448849970116527132072252875, −14.1572063923365341892089314478, −12.90009382895988864962197698261, −12.26239285497704750680752288447, −10.662921882877970703941722535307, −9.8896874197696269684953767700, −8.87696746786090862940974076824, −7.78378018814987836222029950458, −6.33906954874436340643801131498, −4.5748889786731700712029143468, −3.4944502952794582658013976029, −2.63822307847772591806159192359, −1.03031229659419799016585220498, 2.254241742221798356251317370884, 3.66265133425588084152624024599, 4.72393587817105196700777383123, 6.14875352355499406354033835660, 7.26803755527379303305616645457, 8.235941777499881201291585351382, 9.41215757966504376975363377645, 9.85893814734015785719010698935, 12.18587238676675151833481695823, 12.89213709094953298527516425335, 14.13423247510318828530818379327, 14.90897105789964454434445237331, 15.57447788637188052351356950499, 16.67022343039806032709888976171, 17.736044207224068121917199082342, 19.0424970512204899537176234292, 19.533981535381483671474535149818, 21.1228225806359264204268659226, 21.86876959210503315495938092081, 22.816706363571186661257058531220, 24.0133960741527908043251768789, 24.94671316307214105714581316343, 25.633875046989560954496676662541, 26.174295855840590141938459399146, 27.43173639406816280730710508570

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