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Properties

Label 1-2205-2205.1312-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.644 + 0.764i$
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  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.974 + 0.222i)8-s + (0.0747 − 0.997i)11-s + (0.997 + 0.0747i)13-s + (−0.222 − 0.974i)16-s + (−0.930 − 0.365i)17-s + (−0.5 + 0.866i)19-s + (−0.930 + 0.365i)22-s + (−0.149 + 0.988i)23-s + (−0.365 − 0.930i)26-s + (−0.365 + 0.930i)29-s − 31-s + (−0.781 + 0.623i)32-s + (0.0747 + 0.997i)34-s + ⋯
L(s)  = 1  + (−0.433 − 0.900i)2-s + (−0.623 + 0.781i)4-s + (0.974 + 0.222i)8-s + (0.0747 − 0.997i)11-s + (0.997 + 0.0747i)13-s + (−0.222 − 0.974i)16-s + (−0.930 − 0.365i)17-s + (−0.5 + 0.866i)19-s + (−0.930 + 0.365i)22-s + (−0.149 + 0.988i)23-s + (−0.365 − 0.930i)26-s + (−0.365 + 0.930i)29-s − 31-s + (−0.781 + 0.623i)32-s + (0.0747 + 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6151166176 + 0.2860545872i\)
\(L(\frac12)\) \(\approx\) \(0.6151166176 + 0.2860545872i\)
\(L(1)\) \(\approx\) \(0.7060192265 - 0.1945139156i\)
\(L(1)\) \(\approx\) \(0.7060192265 - 0.1945139156i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.433 - 0.900i)T \)
11 \( 1 + (0.0747 - 0.997i)T \)
13 \( 1 + (0.997 + 0.0747i)T \)
17 \( 1 + (-0.930 - 0.365i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.149 + 0.988i)T \)
29 \( 1 + (-0.365 + 0.930i)T \)
31 \( 1 - T \)
37 \( 1 + (-0.149 - 0.988i)T \)
41 \( 1 + (0.733 + 0.680i)T \)
43 \( 1 + (0.680 + 0.733i)T \)
47 \( 1 + (0.433 + 0.900i)T \)
53 \( 1 + (-0.149 + 0.988i)T \)
59 \( 1 + (-0.222 - 0.974i)T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (-0.997 + 0.0747i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.997 + 0.0747i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
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

−19.55376052762620706959142458358, −18.67979187424851658908334036573, −18.096698318204793101510609594547, −17.42741181444419228794709507944, −16.853820927025584726669944969869, −15.9406872637477910569550791834, −15.32655993772234002516547813897, −14.86969594689485860777587662566, −13.898954231338286830740915457513, −13.22735506885725810661747173285, −12.58087585083190750914807505667, −11.376319454759220230624246436222, −10.648425298835437897955715904572, −9.992542122866068783022577419177, −8.937779638290515954773768484693, −8.67943411283252674114213681515, −7.629289977962234687906541879640, −6.90524286335092491582351609363, −6.28325149496587220085139765905, −5.45513714895808178974920008132, −4.439489348497845595747686100417, −3.99323745592440797458512797835, −2.42595031178087129705475301531, −1.5859653180098594031323900141, −0.29025237734123841778877425395, 1.0893890900886056308573066626, 1.84912771127300656051977699472, 2.93056878996676393775694682889, 3.66909713909968167554988744258, 4.32756574198167662859016641760, 5.50318591209983947032969381815, 6.25610621567835277521999802980, 7.41394669553375834629077198878, 8.10559030394186336228106596257, 9.02344512713685920363343831558, 9.28305356373370343536025548221, 10.533173866677182925142076449543, 11.04304518653316865886368361009, 11.50043177875972077850200958110, 12.60130447867255028081395434629, 13.0875455977277689583894398678, 13.91782450766755714683544501845, 14.47935321075531471089471710471, 15.869969929351967412043348591811, 16.21001130388232397327336965975, 17.12390958425733241419744615690, 17.85507247805450053077875842783, 18.53276518956145112002810840766, 19.04603150502193597680279859276, 19.89459710138455288183298238138

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