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Properties

Label 1-185-185.84-r0-0-0
Degree $1$
Conductor $185$
Sign $0.989 + 0.146i$
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 + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + 6-s + (0.5 − 0.866i)7-s − 8-s + (−0.5 − 0.866i)9-s + 11-s + (0.5 + 0.866i)12-s + (0.5 − 0.866i)13-s + 14-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(185\)    =    \(5 \cdot 37\)
Sign: $0.989 + 0.146i$
Analytic conductor: \(0.859136\)
Root analytic conductor: \(0.859136\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{185} (84, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 185,\ (0:\ ),\ 0.989 + 0.146i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.702033464 + 0.1254126183i\)
\(L(\frac12)\) \(\approx\) \(1.702033464 + 0.1254126183i\)
\(L(1)\) \(\approx\) \(1.507527745 + 0.1821017226i\)
\(L(1)\) \(\approx\) \(1.507527745 + 0.1821017226i\)

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

−27.60388540250650980742254007732, −26.44308857699347374942006622692, −25.24671685478114539420224227461, −24.315167222352474304242662947657, −23.068289868910321049896634294068, −22.013228581721399475144718327305, −21.486848247244692962189185947567, −20.664348454008886533701196414011, −19.66093414696688923397476292280, −18.86397853758001236930707694762, −17.66789873692879982659758722052, −16.18794557570385903717365843196, −15.18799872183317969590322671115, −14.30467547735306003203453175297, −13.62527632701931374458932585165, −11.94612721090948557760751803714, −11.467373928444306553375914838273, −10.16304783268726934643677184533, −9.18337656873534024999112632682, −8.48796833718866774248952604171, −6.36434392654950458235586062216, −5.037073156462614083561782061873, −4.19532900827566200343217661069, −2.97164868000308392516364351015, −1.81223489548653750515351876897, 1.37727858296318616570241455100, 3.29302316214225791655031664153, 4.2504514158167553772184910392, 5.948069925076772901157372874477, 6.72471948224281402971426144084, 8.017140058566335578019757346853, 8.37137974366604994558023956682, 10.05176443214779254377092545588, 11.73852630007923337147584208714, 12.65215627408378675734905074695, 13.6825737069693807628285928410, 14.35219811523082985326069712073, 15.18339231403320462183155591109, 16.66611151341352274235132605161, 17.43553986341678199874609899235, 18.26978439621610670775176737910, 19.57594728176509500370819087728, 20.50761991150139907060316595300, 21.55882349769026327196977884732, 22.98649778663084207034998058089, 23.42766609899855429515570166456, 24.48877325757474730769236405047, 25.14470735763668068859913360207, 26.000546179149819179692946957331, 26.95081321895478660784268020368

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