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Properties

Label 1-2205-2205.139-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.225 + 0.974i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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 + (−0.733 − 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.988 + 0.149i)16-s + (−0.900 + 0.433i)17-s − 19-s + (−0.0747 − 0.997i)22-s + (−0.0747 − 0.997i)23-s + (0.900 + 0.433i)26-s + (0.0747 − 0.997i)29-s + (0.5 + 0.866i)31-s + (−0.826 − 0.563i)32-s + (−0.955 − 0.294i)34-s + ⋯
L(s)  = 1  + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)8-s + (−0.733 − 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.988 + 0.149i)16-s + (−0.900 + 0.433i)17-s − 19-s + (−0.0747 − 0.997i)22-s + (−0.0747 − 0.997i)23-s + (0.900 + 0.433i)26-s + (0.0747 − 0.997i)29-s + (0.5 + 0.866i)31-s + (−0.826 − 0.563i)32-s + (−0.955 − 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.225 + 0.974i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.225 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.637658508 + 2.061078288i\)
\(L(\frac12)\) \(\approx\) \(1.637658508 + 2.061078288i\)
\(L(1)\) \(\approx\) \(1.277455013 + 0.6167794925i\)
\(L(1)\) \(\approx\) \(1.277455013 + 0.6167794925i\)

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 \( 1 + (-0.733 - 0.680i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.0747 - 0.997i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.900 - 0.433i)T \)
41 \( 1 + (0.988 + 0.149i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
47 \( 1 + (-0.733 - 0.680i)T \)
53 \( 1 + (0.900 + 0.433i)T \)
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 \( 1 + (-0.222 - 0.974i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.955 + 0.294i)T \)
89 \( 1 + (0.222 + 0.974i)T \)
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

−19.46946437726323469412592299805, −18.74125441138815845680236754835, −18.0685620809563709482316821933, −17.39003070138251830439895505613, −16.11482119443209691746079809511, −15.66345199461060571317985425357, −14.92046605141111634347525451331, −14.17281051966508819134403677870, −13.25059754504923728059851762281, −13.002586153528921811381714596035, −12.08962312629687195640541737965, −11.13566031938273140428428993465, −10.88656086277160385140444659955, −9.828503303046611234348846104200, −9.23606049443344484975630262610, −8.26661978124519218407256689175, −7.22357338486036296435676346542, −6.38255835797587431166317687317, −5.698513181233122081855078006763, −4.68360288341061021711842898690, −4.20779631113957523258946382899, −3.188504587010742721235574586769, −2.330711100845661685178683498664, −1.5999234496787251459678179747, −0.45114462089240480227953820526, 0.70342102230576724499119496269, 2.26402144874680199957929425794, 2.89024833721031644429899749821, 4.02609622416472920128794455997, 4.45983662584382939475823143041, 5.6383625413386479950257426439, 6.12391348268996183334805346514, 6.83824924649214297748077303404, 7.91827835457463499754425522039, 8.43404417544709365292371486757, 9.05736450835282039966492841512, 10.51662100125155200632860727125, 10.92310805382346486633307765726, 11.87105603976159430454964458981, 12.7781643317053090003337497763, 13.27390990381192586666497093325, 13.864340811113025552937001678600, 14.82297360719168493010692350439, 15.35685308243871610539662766884, 16.14985721212137103995638556246, 16.57281796772083504595928373278, 17.63151774686130176506461564291, 18.047786901187490882358772149253, 18.99772035763537047579308030574, 19.83079942405666711705689287506

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