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Properties

Label 1-3381-3381.1583-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.178 + 0.984i$
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  + (−0.917 + 0.396i)2-s + (0.685 − 0.728i)4-s + (0.794 − 0.607i)5-s + (−0.339 + 0.940i)8-s + (−0.488 + 0.872i)10-s + (0.262 + 0.965i)11-s + (0.742 + 0.670i)13-s + (−0.0611 − 0.998i)16-s + (0.685 + 0.728i)17-s + (0.142 + 0.989i)19-s + (0.101 − 0.994i)20-s + (−0.623 − 0.781i)22-s + (0.262 − 0.965i)25-s + (−0.947 − 0.320i)26-s + (−0.685 − 0.728i)29-s + ⋯
L(s)  = 1  + (−0.917 + 0.396i)2-s + (0.685 − 0.728i)4-s + (0.794 − 0.607i)5-s + (−0.339 + 0.940i)8-s + (−0.488 + 0.872i)10-s + (0.262 + 0.965i)11-s + (0.742 + 0.670i)13-s + (−0.0611 − 0.998i)16-s + (0.685 + 0.728i)17-s + (0.142 + 0.989i)19-s + (0.101 − 0.994i)20-s + (−0.623 − 0.781i)22-s + (0.262 − 0.965i)25-s + (−0.947 − 0.320i)26-s + (−0.685 − 0.728i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.178 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.178 + 0.984i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1583, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.178 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.057010875 + 0.8828684644i\)
\(L(\frac12)\) \(\approx\) \(1.057010875 + 0.8828684644i\)
\(L(1)\) \(\approx\) \(0.8732149187 + 0.2324060996i\)
\(L(1)\) \(\approx\) \(0.8732149187 + 0.2324060996i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.917 + 0.396i)T \)
5 \( 1 + (0.794 - 0.607i)T \)
11 \( 1 + (0.262 + 0.965i)T \)
13 \( 1 + (0.742 + 0.670i)T \)
17 \( 1 + (0.685 + 0.728i)T \)
19 \( 1 + (0.142 + 0.989i)T \)
29 \( 1 + (-0.685 - 0.728i)T \)
31 \( 1 + (0.841 + 0.540i)T \)
37 \( 1 + (0.452 + 0.891i)T \)
41 \( 1 + (-0.794 + 0.607i)T \)
43 \( 1 + (-0.339 - 0.940i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (-0.818 - 0.574i)T \)
59 \( 1 + (-0.488 + 0.872i)T \)
61 \( 1 + (-0.947 + 0.320i)T \)
67 \( 1 + (0.654 - 0.755i)T \)
71 \( 1 + (0.999 + 0.0407i)T \)
73 \( 1 + (0.557 + 0.830i)T \)
79 \( 1 + (0.959 + 0.281i)T \)
83 \( 1 + (0.0203 + 0.999i)T \)
89 \( 1 + (-0.523 + 0.852i)T \)
97 \( 1 + (-0.415 - 0.909i)T \)
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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

−18.48077841011677035659834867625, −18.235214414069707349437963945144, −17.313391591867548614265825127166, −16.87220303206448774687508879636, −15.99841975168873830079848373977, −15.43724622376116000744920924885, −14.47819732399753265398480563273, −13.67068540015876890089187422346, −13.16813069999190998824344836086, −12.24729504765160621566067860838, −11.307931589886916149959913365262, −10.953711954639789875250009241067, −10.26552142595649963678984343187, −9.40941973158408297096993234311, −9.00958703186402501465527529795, −8.06821899275764158586253419693, −7.3912818236806803404573049799, −6.52799862491814291991577866225, −5.96589167867591483163791316774, −5.08358843610627336921235770772, −3.65473913233569109717607530603, −3.11744905133456686552614676311, −2.42945192550801665793313202068, −1.40025884867266315848679531809, −0.59410823763167678779171329482, 1.185754848867989812501135239412, 1.59408377892601207869832610415, 2.41425914357011307512157032738, 3.698446916130092460485921943966, 4.67403249850246053988653260102, 5.50414253698831424531731687443, 6.230445670553422743859275975192, 6.726243634038440200082230129772, 7.82027654140612062090095244491, 8.33510946883615510903448201668, 9.13435507482208073682051078134, 9.812459862592328948982344667020, 10.17643096731230441297863726131, 11.11114289096920289295751665329, 12.04123143666303768356461093770, 12.515290936187619754775889210347, 13.65713110237168676369012401158, 14.12072022921623164293737681435, 15.02628989081957337546374239747, 15.56217702015967683777414557136, 16.61619647598579578998853651558, 16.8404171501884453003832874811, 17.46407988376507400363368503587, 18.33218467745843456356124729845, 18.70464712979612363109096901149

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