Properties

Label 1-185-185.109-r1-0-0
Degree $1$
Conductor $185$
Sign $-0.814 + 0.580i$
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 + (0.173 + 0.984i)3-s + (0.939 + 0.342i)4-s + i·6-s + (−0.766 + 0.642i)7-s + (0.866 + 0.5i)8-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)12-s + (−0.342 + 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)18-s + (−0.984 + 0.173i)19-s + ⋯
L(s)  = 1  + (0.984 + 0.173i)2-s + (0.173 + 0.984i)3-s + (0.939 + 0.342i)4-s + i·6-s + (−0.766 + 0.642i)7-s + (0.866 + 0.5i)8-s + (−0.939 + 0.342i)9-s + (0.5 − 0.866i)11-s + (−0.173 + 0.984i)12-s + (−0.342 + 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (0.342 + 0.939i)17-s + (−0.984 + 0.173i)18-s + (−0.984 + 0.173i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $-0.814 + 0.580i$
Analytic conductor: \(19.8810\)
Root analytic conductor: \(19.8810\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (1:\ ),\ -0.814 + 0.580i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8970829341 + 2.805016250i\)
\(L(\frac12)\) \(\approx\) \(0.8970829341 + 2.805016250i\)
\(L(1)\) \(\approx\) \(1.447695974 + 1.113817003i\)
\(L(1)\) \(\approx\) \(1.447695974 + 1.113817003i\)

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 \( 1 + (0.173 + 0.984i)T \)
7 \( 1 + (-0.766 + 0.642i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (0.342 + 0.939i)T \)
19 \( 1 + (-0.984 + 0.173i)T \)
23 \( 1 + (-0.866 + 0.5i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.766 - 0.642i)T \)
59 \( 1 + (-0.642 + 0.766i)T \)
61 \( 1 + (-0.342 + 0.939i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (0.642 + 0.766i)T \)
83 \( 1 + (0.939 - 0.342i)T \)
89 \( 1 + (0.642 - 0.766i)T \)
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

−26.21817695360167901518660588537, −25.21462170977032498270253132660, −24.788262782405476256953595801290, −23.383327317189143288463808116801, −23.04410765004108346392408343389, −22.11028644723050391761913418207, −20.60530587260864217869219582555, −19.91676910473610018475558524551, −19.26594637030987750188792299503, −17.870032830600097304159877767669, −16.826342456519782757169801023453, −15.60396299889285144620217865698, −14.45038187339818828597971306809, −13.71303682726066912325280045730, −12.60519106315199820741779192725, −12.23215108727352154326938916902, −10.77896754232196720032014002960, −9.662424807642959633668314675669, −7.87416342168880448909705228621, −6.91994883154045717689401081522, −6.153836958654087993670101611, −4.676383935226897596587985295666, −3.30688794700516601528527867251, −2.230955777452520216160687021136, −0.69242916744907034174542175922, 2.2644084360854249658510517811, 3.49807788343729457880886020483, 4.30538447053003932697664229575, 5.730062856791486108163454534569, 6.40497358742596829157781401942, 8.16671730300596867632892417754, 9.25998692137417228989314832067, 10.49106925911027284942017850861, 11.56920859340584261654100806330, 12.513470569545742909101184582536, 13.81892097219372593881041094487, 14.61654924006205300124261101829, 15.54822300822275113013824475623, 16.396192990754456175818026664132, 17.074272762293552418006607728211, 19.13948934932376108110696218864, 19.73062665658352417665160008735, 21.1242329721649916145958978678, 21.71837075722424613286832012054, 22.28455047052226512766494254454, 23.436022379191434856535943525948, 24.40924089987343055906781779337, 25.63715994118776285231388874528, 26.0302972994679483960632974285, 27.3226182291610674635176357174

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