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Properties

Label 1-2205-2205.1982-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.0590 + 0.998i$
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  + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (0.433 − 0.900i)8-s + (0.988 + 0.149i)11-s + (0.930 − 0.365i)13-s + (0.826 + 0.563i)16-s + (0.974 + 0.222i)17-s − 19-s + (−0.294 + 0.955i)22-s + (−0.294 + 0.955i)23-s + (0.222 + 0.974i)26-s + (0.955 − 0.294i)29-s + (−0.5 + 0.866i)31-s + (−0.680 + 0.733i)32-s + (−0.365 + 0.930i)34-s + ⋯
L(s)  = 1  + (−0.149 + 0.988i)2-s + (−0.955 − 0.294i)4-s + (0.433 − 0.900i)8-s + (0.988 + 0.149i)11-s + (0.930 − 0.365i)13-s + (0.826 + 0.563i)16-s + (0.974 + 0.222i)17-s − 19-s + (−0.294 + 0.955i)22-s + (−0.294 + 0.955i)23-s + (0.222 + 0.974i)26-s + (0.955 − 0.294i)29-s + (−0.5 + 0.866i)31-s + (−0.680 + 0.733i)32-s + (−0.365 + 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0590 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0590 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0590 + 0.998i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1982, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.0590 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100411088 + 1.037254807i\)
\(L(\frac12)\) \(\approx\) \(1.100411088 + 1.037254807i\)
\(L(1)\) \(\approx\) \(0.9081828245 + 0.4783707786i\)
\(L(1)\) \(\approx\) \(0.9081828245 + 0.4783707786i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.149 + 0.988i)T \)
11 \( 1 + (0.988 + 0.149i)T \)
13 \( 1 + (0.930 - 0.365i)T \)
17 \( 1 + (0.974 + 0.222i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.294 + 0.955i)T \)
29 \( 1 + (0.955 - 0.294i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.974 - 0.222i)T \)
41 \( 1 + (-0.826 + 0.563i)T \)
43 \( 1 + (0.563 - 0.826i)T \)
47 \( 1 + (-0.149 + 0.988i)T \)
53 \( 1 + (0.974 - 0.222i)T \)
59 \( 1 + (0.0747 - 0.997i)T \)
61 \( 1 + (0.955 - 0.294i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.781 - 0.623i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.930 + 0.365i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−19.52602977654603869370644988382, −18.904297726539259471671206607968, −18.3647479401307608547015101457, −17.51350256232992180333745039493, −16.75167539385672313867767022301, −16.257513032200684052323062911745, −14.963374294880464821867677840948, −14.33571313626615726185379684881, −13.65852368376718113001348102771, −12.92012597930433624348823529081, −11.985789265545009933045543472310, −11.71197794479165094659909207766, −10.60420712329444314306236886836, −10.25473078915664583435125953875, −9.119228794971316942606559137283, −8.72424969787357853831056302709, −7.92008354618849420243658951079, −6.7756730248058513588097900527, −5.989283503612216544252183487523, −4.96770100185142641218381870247, −4.02954468811575669693493162136, −3.56831486670250169462615945715, −2.47963641019405148214253031716, −1.60343955797374106739285583080, −0.72841312530450820277471568341, 0.9218259864565986067187545652, 1.77438188369942216900761982118, 3.42586032078780425578882861727, 3.9054338339352453208629927842, 4.95333308895012283880728828420, 5.7545094585540797465560566819, 6.44815047720310435564432146353, 7.1105080113397306020302994102, 8.11029780232661208705388633309, 8.598186189436225909500572226112, 9.42568052409018548362718225688, 10.188202443918788400832695911530, 10.945868117844893494545410670122, 12.065301971086424238344509117131, 12.69940510183358875826140352043, 13.6574500305380933409577697255, 14.17340751790618143463912532201, 14.925538051645754680218023206977, 15.61220287608603595595191944181, 16.2819199630243078413158140878, 17.05235169483341160821361994122, 17.5901471836789354826046077132, 18.27852592365038497379742208688, 19.2276324385819660114495930752, 19.52843147002967723522405767002

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