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Properties

Label 1-2205-2205.212-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.988 + 0.150i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (0.733 − 0.680i)11-s + (0.680 + 0.733i)13-s + (0.623 − 0.781i)16-s + (−0.563 − 0.826i)17-s + (0.5 + 0.866i)19-s + (−0.563 + 0.826i)22-s + (0.997 + 0.0747i)23-s + (−0.826 − 0.563i)26-s + (0.826 − 0.563i)29-s + 31-s + (−0.433 + 0.900i)32-s + (0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (0.733 − 0.680i)11-s + (0.680 + 0.733i)13-s + (0.623 − 0.781i)16-s + (−0.563 − 0.826i)17-s + (0.5 + 0.866i)19-s + (−0.563 + 0.826i)22-s + (0.997 + 0.0747i)23-s + (−0.826 − 0.563i)26-s + (0.826 − 0.563i)29-s + 31-s + (−0.433 + 0.900i)32-s + (0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.988 + 0.150i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (212, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.988 + 0.150i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.198333089 + 0.09069469464i\)
\(L(\frac12)\) \(\approx\) \(1.198333089 + 0.09069469464i\)
\(L(1)\) \(\approx\) \(0.8184282528 + 0.05259736495i\)
\(L(1)\) \(\approx\) \(0.8184282528 + 0.05259736495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (0.733 - 0.680i)T \)
13 \( 1 + (0.680 + 0.733i)T \)
17 \( 1 + (-0.563 - 0.826i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (0.997 + 0.0747i)T \)
29 \( 1 + (0.826 - 0.563i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.997 + 0.0747i)T \)
41 \( 1 + (-0.365 + 0.930i)T \)
43 \( 1 + (0.930 - 0.365i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (0.997 + 0.0747i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.680 + 0.733i)T \)
79 \( 1 - T \)
83 \( 1 + (0.680 - 0.733i)T \)
89 \( 1 + (0.955 - 0.294i)T \)
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

−19.58102810945675754409559428180, −19.17419946054925632259945265228, −18.03747097917908527207524997012, −17.68708961216552510947988122107, −17.08434654045996564403494037516, −16.201735323771963148944466936478, −15.39007614034124214500930697801, −15.04729758271135966820980985474, −13.845720009681918835574091255983, −12.98585920657741325505151601209, −12.27658242996820559412990585037, −11.55545332480809884288909338209, −10.71182168230632825645052295618, −10.25964083464156724852676579845, −9.21394749586882672999374085855, −8.75532608374661780632957115181, −7.97568095788129672236453378589, −6.96564609415918093849358730773, −6.58438820804509487728395011097, −5.506189657174856343253154227667, −4.40726145201355395183464962603, −3.43735199323784644199093641793, −2.650798711073451654307118487375, −1.60833988867025553364411333124, −0.82250484288553919841569939791, 0.82847321769729057320296365612, 1.56393581436611287848618228400, 2.683493893510083778286754950158, 3.52619017543797532246312048466, 4.65341158046507196674374281933, 5.68523921387877607018658737443, 6.49854768100117160871404561900, 6.96499212675019498264737792051, 8.03348493259778593376547290741, 8.68011720242745734399736626064, 9.29884693856679093186159359767, 10.04021946461908044316043619875, 10.93717010925931439874737162586, 11.59909810866858368049433166836, 12.05537636194185823742394123510, 13.396659138532072970789149050091, 14.05272321311921971762475645830, 14.74508813106977361294714970466, 15.81679444416584112582282951103, 16.09721552933286342260876490661, 16.96835515203077477332804461137, 17.52076477571525207614559620574, 18.40857831221498885695289286479, 18.96661563049224472338246226024, 19.50039060049936166526945847907

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