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Properties

Label 1-3381-3381.44-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.575 - 0.817i$
Analytic cond. $15.7012$
Root an. cond. $15.7012$
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  + (−0.665 − 0.746i)2-s + (−0.115 + 0.993i)4-s + (−0.751 − 0.659i)5-s + (0.818 − 0.574i)8-s + (0.00679 + 0.999i)10-s + (−0.923 + 0.384i)11-s + (−0.933 − 0.359i)13-s + (−0.973 − 0.229i)16-s + (−0.802 + 0.596i)17-s + (−0.981 − 0.189i)19-s + (0.742 − 0.670i)20-s + (0.900 + 0.433i)22-s + (0.128 + 0.991i)25-s + (0.352 + 0.935i)26-s + (−0.917 − 0.396i)29-s + ⋯
L(s)  = 1  + (−0.665 − 0.746i)2-s + (−0.115 + 0.993i)4-s + (−0.751 − 0.659i)5-s + (0.818 − 0.574i)8-s + (0.00679 + 0.999i)10-s + (−0.923 + 0.384i)11-s + (−0.933 − 0.359i)13-s + (−0.973 − 0.229i)16-s + (−0.802 + 0.596i)17-s + (−0.981 − 0.189i)19-s + (0.742 − 0.670i)20-s + (0.900 + 0.433i)22-s + (0.128 + 0.991i)25-s + (0.352 + 0.935i)26-s + (−0.917 − 0.396i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.575 - 0.817i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.575 - 0.817i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (44, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.575 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2830260367 - 0.1468634454i\)
\(L(\frac12)\) \(\approx\) \(0.2830260367 - 0.1468634454i\)
\(L(1)\) \(\approx\) \(0.4352715816 - 0.1657841380i\)
\(L(1)\) \(\approx\) \(0.4352715816 - 0.1657841380i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.665 - 0.746i)T \)
5 \( 1 + (-0.751 - 0.659i)T \)
11 \( 1 + (-0.923 + 0.384i)T \)
13 \( 1 + (-0.933 - 0.359i)T \)
17 \( 1 + (-0.802 + 0.596i)T \)
19 \( 1 + (-0.981 - 0.189i)T \)
29 \( 1 + (-0.917 - 0.396i)T \)
31 \( 1 + (0.723 - 0.690i)T \)
37 \( 1 + (-0.999 - 0.0271i)T \)
41 \( 1 + (-0.947 + 0.320i)T \)
43 \( 1 + (0.818 + 0.574i)T \)
47 \( 1 + (-0.955 + 0.294i)T \)
53 \( 1 + (-0.675 - 0.737i)T \)
59 \( 1 + (0.00679 + 0.999i)T \)
61 \( 1 + (0.634 + 0.773i)T \)
67 \( 1 + (0.995 + 0.0950i)T \)
71 \( 1 + (-0.0203 + 0.999i)T \)
73 \( 1 + (-0.848 + 0.529i)T \)
79 \( 1 + (-0.928 + 0.371i)T \)
83 \( 1 + (-0.714 - 0.699i)T \)
89 \( 1 + (0.511 - 0.859i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
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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

−18.91301503944942035927091720385, −18.29740058725313938222380292367, −17.49810483875677845914586875603, −16.914652865546947635142057112741, −15.96923414552598921924353323978, −15.65312620208131915440575437661, −14.92637224907688012262509027082, −14.29820216209292623417648973236, −13.62635635927291297307293888355, −12.66738158344769756492216859506, −11.7722896088623669210106421040, −10.95487618598231288962667662424, −10.53176359399613652154004986894, −9.744660540375987297657113765703, −8.839936799132084209732631558884, −8.22765968752030525949140385658, −7.50689785973064666267351905796, −6.89370208767235732742718700524, −6.33215935051861930024435931837, −5.19596805919575794518917841877, −4.70389531812524354299429780860, −3.64472149973615164191321759195, −2.59579121460210650118357213617, −1.87837973971151507948447270550, −0.32957118366922863548595420598, 0.33619546398858710636181838110, 1.6286744487253349830325879575, 2.37299471434133238335193076655, 3.18944828563233032817698999477, 4.27302008228092362875565009706, 4.58627575094092854110996325077, 5.62698696597107709287600382699, 6.91468525492749867052797712370, 7.51219447582159344933092876919, 8.3233887368743745758519432614, 8.63122076213999890455107774391, 9.72341547150845803937979702054, 10.17204952711473535694519279639, 11.116458988571587226788640682195, 11.55864139495005912124600550912, 12.5686342497361946654794990457, 12.82026999544305865831062009801, 13.44395023961445532358375050425, 14.76807522485390110381153661198, 15.4137865219777302896051472293, 15.99863031892021718007562971349, 16.9245688070640405224116244896, 17.33711988433779803902736848744, 17.99414366507716856014953342046, 19.08000219070710009202961269801

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