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Properties

Label 1-3724-3724.1475-r1-0-0
Degree $1$
Conductor $3724$
Sign $0.0444 + 0.999i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (0.988 + 0.149i)3-s + (−0.623 + 0.781i)5-s + (0.955 + 0.294i)9-s + (0.733 + 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.733 + 0.680i)15-s + (−0.826 − 0.563i)17-s + (−0.826 + 0.563i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.826 − 0.563i)29-s + (0.5 − 0.866i)31-s + (0.623 + 0.781i)33-s + (−0.826 − 0.563i)37-s + (0.988 − 0.149i)39-s + ⋯
L(s)  = 1  + (0.988 + 0.149i)3-s + (−0.623 + 0.781i)5-s + (0.955 + 0.294i)9-s + (0.733 + 0.680i)11-s + (0.955 − 0.294i)13-s + (−0.733 + 0.680i)15-s + (−0.826 − 0.563i)17-s + (−0.826 + 0.563i)23-s + (−0.222 − 0.974i)25-s + (0.900 + 0.433i)27-s + (−0.826 − 0.563i)29-s + (0.5 − 0.866i)31-s + (0.623 + 0.781i)33-s + (−0.826 − 0.563i)37-s + (0.988 − 0.149i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0444 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0444 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.0444 + 0.999i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ 0.0444 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.306574342 + 2.206229460i\)
\(L(\frac12)\) \(\approx\) \(2.306574342 + 2.206229460i\)
\(L(1)\) \(\approx\) \(1.406164741 + 0.3783943673i\)
\(L(1)\) \(\approx\) \(1.406164741 + 0.3783943673i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.988 + 0.149i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (0.733 + 0.680i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.826 - 0.563i)T \)
23 \( 1 + (-0.826 + 0.563i)T \)
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 \( 1 + (0.988 - 0.149i)T \)
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 \( 1 + T \)
71 \( 1 + (0.826 - 0.563i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.222 - 0.974i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
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

−18.51323584613529377795247145738, −17.61870131007499154008828176766, −16.82874402442189622727098809578, −15.94630001698360108314733836509, −15.76048059423482889834371394657, −14.82816641439925268776075117313, −14.0335297274692406220268748479, −13.636873604568465793896017128217, −12.68879182871788516099022627715, −12.34321255285476657614969130765, −11.30687655232421241436927908662, −10.77188513703159115529682364034, −9.6242240557093197192448164927, −9.00099994151946108826306565619, −8.398738903733450480925031422426, −8.108046076689637474005400133521, −6.94471471602778409709850870042, −6.42539438879163318720266229125, −5.37209651528801365225256195072, −4.312537191600673120603733006989, −3.85816473703608392017782966442, −3.2368008386532543456200105285, −2.01724033202533403424741966550, −1.3696858768023602959320758293, −0.47822285055385519701505983403, 0.8369322112175878882811420649, 1.99825801235875188468224911402, 2.538720400316453502162493595334, 3.68432874345184757916994098832, 3.868988770309844078568286284747, 4.74739741337867074670060511429, 6.01503167506369638499003599810, 6.72251972301284862299038545894, 7.49726452240239146244740822564, 7.95399460289855273667121270874, 8.86858558570871520873281815467, 9.45688504268425114935756786530, 10.214153630216464705495014704602, 10.9684219488434393820630428756, 11.65993082674128537621644481310, 12.37973180707946980336145965895, 13.474262790391332047869987774057, 13.703941311600300990246350402487, 14.70426148914860744552648172146, 15.090971472054769110641258878176, 15.73156457737070288557918412612, 16.271501606661776071225096443630, 17.408420607559104901037876935438, 18.16157019078183902204134129641, 18.59672690210830517488713452893

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