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Properties

Label 1-2205-2205.1024-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.992 + 0.120i$
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  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)8-s + (−0.222 + 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 + 0.563i)22-s + (0.900 + 0.433i)23-s + (0.0747 + 0.997i)26-s + (0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.826 − 0.563i)32-s + (−0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)8-s + (−0.222 + 0.974i)11-s + (0.733 + 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 + 0.563i)22-s + (0.900 + 0.433i)23-s + (0.0747 + 0.997i)26-s + (0.826 + 0.563i)29-s + (−0.5 − 0.866i)31-s + (−0.826 − 0.563i)32-s + (−0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.992 + 0.120i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1024, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.992 + 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1199960463 + 1.979272007i\)
\(L(\frac12)\) \(\approx\) \(0.1199960463 + 1.979272007i\)
\(L(1)\) \(\approx\) \(1.100473920 + 0.9328408580i\)
\(L(1)\) \(\approx\) \(1.100473920 + 0.9328408580i\)

Euler product

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

−19.39253016863748013614604981601, −18.7872305316490760786214035870, −18.26622338920689130386377276895, −17.2824694230593073673111877497, −16.22839803881703263762981781132, −15.743685249454087572684013575557, −14.9240285745646451000147695878, −14.06947373595397016227610670124, −13.58635660245360798639740981973, −12.86406007892030500484216849018, −12.08218929796229616866716044285, −11.40438513605278634161099701640, −10.5274040045292848748604546681, −10.26052805765925602709302654200, −8.95278128908060662296998785537, −8.51245532265115306473512735130, −7.27229412406828827144901723326, −6.40255898276799843498381518036, −5.62472260855186025522431076793, −5.0309260492798317636381121745, −3.97913226763074624428242092212, −3.247059825807408628894924326888, −2.58352129380002305181938911039, −1.430472085224858953824625904814, −0.50000068487290338607138654090, 1.51304833863710227958015568663, 2.47435031262630858483283648707, 3.43319398118283174402540303704, 4.32617365713175036284682880709, 4.84127780464647156341361528346, 5.86309096320295767031372352214, 6.59006848854540241631569102954, 7.20557223394957894025885172792, 8.07753694688653754178397734488, 8.846574389385699072142487599874, 9.55987962876292392169976987846, 10.80987620392709665197435317882, 11.320288115408838471738678715894, 12.35666613936685815638721565162, 12.91224094678390220691564028992, 13.491945500810658556591686687799, 14.451004452543059276660687635795, 14.96382302731844940219997234553, 15.69856576251433948074046865000, 16.271946220690891534156251175674, 17.34011901139729555964824806276, 17.50869679921260971688962119776, 18.539178698744370505944125637314, 19.391804247106913866318485063932, 20.263254503822432577840551877161

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