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Properties

Label 1-185-185.174-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: $\chi_{185} (174, \cdot )$
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

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

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