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Properties

Label 1-185-185.77-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.866 - 0.499i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.866 - 0.499i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.866 - 0.499i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2909040746 - 1.086407662i\)
\(L(\frac12)\) \(\approx\) \(0.2909040746 - 1.086407662i\)
\(L(1)\) \(\approx\) \(1.057760853 - 0.3684891841i\)
\(L(1)\) \(\approx\) \(1.057760853 - 0.3684891841i\)

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

−27.38205866794001637921076210395, −26.29486939155764823678644093124, −25.38033108869489386624903960886, −24.03909995185673905529350122382, −23.500344405365748915249904719, −22.61330100262249430051034239255, −22.00132148673689014777136080944, −20.81165682295479646739499476659, −20.04825921232351809697614097144, −18.58203108463388060394159463595, −17.344630479895055529088734868041, −16.398914826567522041318018899692, −15.873168572290311973359026022125, −14.56901769527110556811076171037, −13.46787846874616620394068949567, −12.501407411281436062342359007777, −11.69932477525400839265366145414, −10.57076600432056388405121962414, −9.653794794849273939372221943755, −7.40866373832858619957456392927, −6.85307840796494004884082202469, −5.614754019931103049805596526056, −4.58910025714515911987897933049, −3.62948607506872931453068111410, −1.81040952614667380593203180056, 0.29709161892293895924452696448, 2.16716377028459915771239964517, 3.483035074403805509065681785731, 4.98947764374517235985723583336, 5.83070503588483482626604459254, 6.59431328214339881187579627893, 8.03317798675461602997612601182, 9.88253418440121408647141881353, 10.87754848401857622863661633788, 11.753268310576973465222633375089, 12.84451660626450989827491538097, 13.25602505392624050043435347855, 14.92837434526177186817089335712, 15.77444577886358275879217094026, 16.54179350628714314138038704431, 17.85345915218275227037466782460, 18.94710465320323777483831015225, 19.85128321564882256435065774359, 21.25445355333136371013171069176, 22.089864589532331370042726718915, 22.48621520387662136247271880505, 23.74373754191063350293354999951, 24.30008489889106368460548405795, 25.29086785295084965439212928241, 26.53177894787246926282779544358

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