Properties

Label 1-185-185.9-r0-0-0
Degree $1$
Conductor $185$
Sign $0.883 - 0.468i$
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  + (0.939 − 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)12-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.939 + 0.342i)18-s + (−0.939 − 0.342i)19-s + ⋯
L(s)  = 1  + (0.939 − 0.342i)2-s + (0.939 + 0.342i)3-s + (0.766 − 0.642i)4-s + 6-s + (−0.173 − 0.984i)7-s + (0.5 − 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 + 0.866i)11-s + (0.939 − 0.342i)12-s + (−0.766 + 0.642i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)16-s + (−0.766 − 0.642i)17-s + (0.939 + 0.342i)18-s + (−0.939 − 0.342i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.883 - 0.468i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.883 - 0.468i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.883 - 0.468i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.402278691 - 0.5979289452i\)
\(L(\frac12)\) \(\approx\) \(2.402278691 - 0.5979289452i\)
\(L(1)\) \(\approx\) \(2.091884467 - 0.3675858007i\)
\(L(1)\) \(\approx\) \(2.091884467 - 0.3675858007i\)

Euler product

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

−26.84735345203757100528943741289, −26.14944341498812569258057578074, −24.97641482820052694387214612945, −24.66733254255554576524818317583, −23.67694071705839862497725905623, −22.44095055471852501708471172406, −21.517814074484874124061920711485, −20.832064594377349558730494825015, −19.62351553865699539889566749676, −18.84901002368527095098925248927, −17.53231443592295565557746374970, −16.1997525646825406916826175183, −15.11987483970539187560434611014, −14.7565614636768930065449583273, −13.33187265745503476631745719604, −12.84234960980864560173369768819, −11.767499557321628536722274567720, −10.31992390344944262994141374629, −8.66952142849258077889815956651, −8.07591320034656261172697901174, −6.69956573679099638333420097154, −5.71845780182986480272105669289, −4.31384696525372621672174598638, −2.95039963479978361839894309927, −2.24700451696187811317899161108, 1.85944921225944632917001584369, 2.94657147940939088929702382022, 4.2464997410042108158203965983, 4.85917552376657894539692425593, 6.83357076035440828438900680674, 7.504852304517822226611831594266, 9.27810962562250824488533137294, 10.174631481198949279568874778933, 11.13074954334971553065752115973, 12.61956503244951972518909926885, 13.42010486430302576477648580829, 14.25091472421028831023415849065, 15.16785160197026986274794368303, 16.005873483323317095982485442005, 17.26831833379825397891926519161, 18.95099674866401329646924140080, 19.789334067501172418304132060994, 20.44761384073971520804667436055, 21.29532308844993650865861805624, 22.24789233004131336629505996240, 23.31886811041480395967792992390, 24.142715137122035695882593523972, 25.21605987114781439089530189427, 26.07448179550257105169996792684, 27.02291053883934229937500801753

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