↑

Properties

Label 1-3381-3381.389-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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.314 + 0.949i)2-s + (−0.802 − 0.596i)4-s + (−0.195 + 0.980i)5-s + (0.818 − 0.574i)8-s + (−0.869 − 0.494i)10-s + (0.128 − 0.991i)11-s + (−0.933 − 0.359i)13-s + (0.288 + 0.957i)16-s + (−0.115 − 0.993i)17-s + (0.327 + 0.945i)19-s + (0.742 − 0.670i)20-s + (0.900 + 0.433i)22-s + (−0.923 − 0.384i)25-s + (0.634 − 0.773i)26-s + (−0.917 − 0.396i)29-s + ⋯
L(s)  = 1  + (−0.314 + 0.949i)2-s + (−0.802 − 0.596i)4-s + (−0.195 + 0.980i)5-s + (0.818 − 0.574i)8-s + (−0.869 − 0.494i)10-s + (0.128 − 0.991i)11-s + (−0.933 − 0.359i)13-s + (0.288 + 0.957i)16-s + (−0.115 − 0.993i)17-s + (0.327 + 0.945i)19-s + (0.742 − 0.670i)20-s + (0.900 + 0.433i)22-s + (−0.923 − 0.384i)25-s + (0.634 − 0.773i)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: $\chi_{3381} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.575 - 0.817i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3476324291 - 0.1803879844i\)
\(L(\frac12)\) \(\approx\) \(0.3476324291 - 0.1803879844i\)
\(L(1)\) \(\approx\) \(0.6168110140 + 0.2921814506i\)
\(L(1)\) \(\approx\) \(0.6168110140 + 0.2921814506i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.314 + 0.949i)T \)
5 \( 1 + (-0.195 + 0.980i)T \)
11 \( 1 + (0.128 - 0.991i)T \)
13 \( 1 + (-0.933 - 0.359i)T \)
17 \( 1 + (-0.115 - 0.993i)T \)
19 \( 1 + (0.327 + 0.945i)T \)
29 \( 1 + (-0.917 - 0.396i)T \)
31 \( 1 + (0.235 + 0.971i)T \)
37 \( 1 + (0.476 + 0.879i)T \)
41 \( 1 + (-0.947 + 0.320i)T \)
43 \( 1 + (0.818 + 0.574i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (0.976 - 0.215i)T \)
59 \( 1 + (-0.869 - 0.494i)T \)
61 \( 1 + (0.352 - 0.935i)T \)
67 \( 1 + (-0.580 + 0.814i)T \)
71 \( 1 + (-0.0203 + 0.999i)T \)
73 \( 1 + (-0.0339 - 0.999i)T \)
79 \( 1 + (0.786 + 0.618i)T \)
83 \( 1 + (-0.714 - 0.699i)T \)
89 \( 1 + (-0.999 - 0.0135i)T \)
97 \( 1 + (-0.841 - 0.540i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.14310173904758373771072403173, −18.26320550891903512436546880604, −17.503605437954249249256625389040, −17.01313629104963981063118918507, −16.52089626434293828918847551249, −15.3608246326687302956214676519, −14.857449975141106471005790097414, −13.75665185218308154189098837087, −13.18115389190586405338011659986, −12.41350919853256575355102494473, −12.11444442829339512431630843552, −11.297169781212100215455333183513, −10.48140464950639454629159901780, −9.6730792367229908510278442040, −9.17962436336706139019573461643, −8.57173741776480430459226719838, −7.582597425515336209787346265978, −7.16268132995811430849112382538, −5.72875762863033210096574710706, −4.952003921531185718859418392663, −4.285463628970561092133442138293, −3.751706195553021079584894863102, −2.437885038778432455652318041304, −1.93957383012149906656880400167, −0.96722580290700447608425426915, 0.1537545899019793178826731669, 1.34998142491725667987858765209, 2.6433178863480535103621655631, 3.36824317418307427062682740487, 4.287269812082457804360717595725, 5.23620780031543433415161924065, 5.89596554245742221615408376611, 6.62537837503613069568953683625, 7.355987466466557603669876784630, 7.86919406903031135806408823139, 8.64408657462869324656626456332, 9.61689246368105096441233923271, 10.05483004024437337736656591387, 10.90998357768434602253288698121, 11.5924487167132956585112870581, 12.491092030471923422190150120981, 13.535775425290762084274219048292, 14.0537076445861102802075818526, 14.60697939273945498480229689784, 15.283800115012836736458570672325, 15.95270827870299990780690911067, 16.61679840234486647190914141490, 17.27537259503112680072349771754, 18.08794303232595232260622319984, 18.59969606168273921264371727911

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