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Properties

Label 1-47e2-2209.1035-r0-0-0
Degree $1$
Conductor $2209$
Sign $0.946 + 0.323i$
Analytic cond. $10.2585$
Root an. cond. $10.2585$
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  + (−0.824 − 0.565i)2-s + (0.593 + 0.805i)3-s + (0.359 + 0.933i)4-s + (−0.979 − 0.199i)5-s + (−0.0334 − 0.999i)6-s + (−0.420 − 0.907i)7-s + (0.231 − 0.972i)8-s + (−0.296 + 0.955i)9-s + (0.695 + 0.718i)10-s + (−0.944 + 0.328i)11-s + (−0.538 + 0.842i)12-s + (−0.944 + 0.328i)13-s + (−0.166 + 0.986i)14-s + (−0.420 − 0.907i)15-s + (−0.741 + 0.670i)16-s + (−0.892 + 0.451i)17-s + ⋯
L(s)  = 1  + (−0.824 − 0.565i)2-s + (0.593 + 0.805i)3-s + (0.359 + 0.933i)4-s + (−0.979 − 0.199i)5-s + (−0.0334 − 0.999i)6-s + (−0.420 − 0.907i)7-s + (0.231 − 0.972i)8-s + (−0.296 + 0.955i)9-s + (0.695 + 0.718i)10-s + (−0.944 + 0.328i)11-s + (−0.538 + 0.842i)12-s + (−0.944 + 0.328i)13-s + (−0.166 + 0.986i)14-s + (−0.420 − 0.907i)15-s + (−0.741 + 0.670i)16-s + (−0.892 + 0.451i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2209 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2209\)    =    \(47^{2}\)
Sign: $0.946 + 0.323i$
Analytic conductor: \(10.2585\)
Root analytic conductor: \(10.2585\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2209} (1035, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2209,\ (0:\ ),\ 0.946 + 0.323i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4624864668 + 0.07699744385i\)
\(L(\frac12)\) \(\approx\) \(0.4624864668 + 0.07699744385i\)
\(L(1)\) \(\approx\) \(0.5373659266 + 0.01381403600i\)
\(L(1)\) \(\approx\) \(0.5373659266 + 0.01381403600i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad47 \( 1 \)
good2 \( 1 + (-0.824 - 0.565i)T \)
3 \( 1 + (0.593 + 0.805i)T \)
5 \( 1 + (-0.979 - 0.199i)T \)
7 \( 1 + (-0.420 - 0.907i)T \)
11 \( 1 + (-0.944 + 0.328i)T \)
13 \( 1 + (-0.944 + 0.328i)T \)
17 \( 1 + (-0.892 + 0.451i)T \)
19 \( 1 + (-0.979 - 0.199i)T \)
23 \( 1 + (0.695 - 0.718i)T \)
29 \( 1 + (0.359 - 0.933i)T \)
31 \( 1 + (0.231 - 0.972i)T \)
37 \( 1 + (-0.892 - 0.451i)T \)
41 \( 1 + (-0.997 + 0.0667i)T \)
43 \( 1 + (0.359 - 0.933i)T \)
53 \( 1 + T \)
59 \( 1 + (-0.979 + 0.199i)T \)
61 \( 1 + (0.964 - 0.264i)T \)
67 \( 1 + T \)
71 \( 1 + T \)
73 \( 1 + (-0.0334 + 0.999i)T \)
79 \( 1 + (-0.166 + 0.986i)T \)
83 \( 1 + (-0.979 - 0.199i)T \)
89 \( 1 + (-0.420 + 0.907i)T \)
97 \( 1 + (0.480 + 0.876i)T \)
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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

−19.48938546823110547870425538549, −18.97164071940980154469941968176, −18.341427955934665845604022960062, −17.78517808636587643483948848984, −16.86763311636544726612776595021, −15.85637164437763077323615591261, −15.422342111794193781081396784017, −14.90726476554371412363118576742, −14.09939065753183968405763472719, −13.08388481029767391817708554379, −12.40679425851716279504508875107, −11.65768145234092603146777963471, −10.82777372755203255573789961176, −9.9636936308679694542319842980, −8.93423850027325749791370704689, −8.533371933126929433201911229748, −7.87060620324358554322613027943, −7.00921848430251040806963348761, −6.66092984196134948467621462099, −5.519712900310503759634765551, −4.763057231185210078789560169708, −3.18031862321331842371184283142, −2.697448171538703167612632354123, −1.75956978391930400782324977834, −0.37972875474528525233918111837, 0.48848053013243017286485653383, 2.172007598390024620276124120, 2.68469929301112868241126783316, 3.82487197563384276041366098817, 4.21262629544795271096761038752, 4.98285842033458249102441218932, 6.73491814417186692356188513775, 7.305750791765298364882525994772, 8.1494064261143042391278188691, 8.612138486171907920819831459991, 9.53135374697209173943562117915, 10.250532302339993389626879156422, 10.749900656509317356735974070717, 11.41467773755811226637039279851, 12.49797936831523285681734399388, 13.02966622452786682748372774047, 13.87421486176373314316299989893, 15.14260645876409729045859616680, 15.41798842103639230337614835716, 16.234032912255988071791085448709, 17.02805147863141260484352049091, 17.28131538464289442311856100294, 18.76216658387458077160755371708, 19.14828976365697259854477818389, 19.85948575042744632991953227259

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