Properties

Label 1-185-185.88-r0-0-0
Degree $1$
Conductor $185$
Sign $0.956 + 0.292i$
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  + (−0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s i·6-s + (0.866 + 0.5i)7-s + 8-s + (0.5 + 0.866i)9-s − 11-s + (−0.866 + 0.5i)12-s + (−0.5 + 0.866i)13-s i·14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s i·6-s + (0.866 + 0.5i)7-s + 8-s + (0.5 + 0.866i)9-s − 11-s + (−0.866 + 0.5i)12-s + (−0.5 + 0.866i)13-s i·14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.292i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.956 + 0.292i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (88, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.956 + 0.292i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.143936158 + 0.1707551985i\)
\(L(\frac12)\) \(\approx\) \(1.143936158 + 0.1707551985i\)
\(L(1)\) \(\approx\) \(1.070250697 + 0.01977063608i\)
\(L(1)\) \(\approx\) \(1.070250697 + 0.01977063608i\)

Euler product

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

−27.08555706111500004271304141097, −26.083740970642334245845610139489, −25.28599765010391753946487395313, −24.47419575666070928618298388611, −23.67946419376025656348440327337, −22.85793367126899518691889838838, −21.08558294013987621214295826848, −20.33672241582305440401951340472, −19.24430766187661830630982719441, −18.329632531546609092001756973191, −17.63808963565696345201538576696, −16.48511820820594447897703720836, −15.18737259516915248287079266640, −14.6228097104482401759954085225, −13.61479302231860344589656473044, −12.68178701325670011733090098660, −10.88234807060005411072985346463, −9.88002649194558852444328238058, −8.66059180116940685445249083856, −7.7424200098522042578949201716, −7.24226443212672489247618374411, −5.64392218132957421089327902098, −4.45860355095487107433315400388, −2.65575820101547528279647412, −1.08990743883525153779600839099, 1.86759387521089751040814822580, 2.70393937214537821153822928356, 4.127675226152761324519338107922, 5.09182234484778669187827321958, 7.3854090395548158594346925540, 8.37037360605351078443009411525, 9.06715697225477067082634342377, 10.27080282485480884337430880878, 11.04642443668160554448259720803, 12.32615035490301768392629139845, 13.3618405478907828934498628084, 14.46320849372813840169652879181, 15.42415258014013427369164074215, 16.70633699230286969122104325504, 17.73770125709401753589352988388, 18.95863092786528455686499265446, 19.38854096426754306231687892870, 20.78904866796952595437445061696, 21.22437654742787140445566292323, 21.86801402154754768397687101175, 23.398814531903199798196932543533, 24.62174928354174131488419213716, 25.74557947600639328465264180662, 26.42185430523876243535052452051, 27.29045324650021072582669995215

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