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Properties

Label 1-3381-3381.1223-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.599 - 0.800i$
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  + (0.966 − 0.255i)2-s + (0.869 − 0.494i)4-s + (0.665 + 0.746i)5-s + (0.714 − 0.699i)8-s + (0.833 + 0.552i)10-s + (0.802 − 0.596i)11-s + (−0.986 + 0.162i)13-s + (0.511 − 0.859i)16-s + (−0.00679 − 0.999i)17-s + (0.786 − 0.618i)19-s + (0.947 + 0.320i)20-s + (0.623 − 0.781i)22-s + (−0.115 + 0.993i)25-s + (−0.912 + 0.409i)26-s + (0.862 − 0.505i)29-s + ⋯
L(s)  = 1  + (0.966 − 0.255i)2-s + (0.869 − 0.494i)4-s + (0.665 + 0.746i)5-s + (0.714 − 0.699i)8-s + (0.833 + 0.552i)10-s + (0.802 − 0.596i)11-s + (−0.986 + 0.162i)13-s + (0.511 − 0.859i)16-s + (−0.00679 − 0.999i)17-s + (0.786 − 0.618i)19-s + (0.947 + 0.320i)20-s + (0.623 − 0.781i)22-s + (−0.115 + 0.993i)25-s + (−0.912 + 0.409i)26-s + (0.862 − 0.505i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.599 - 0.800i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.599 - 0.800i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.599 - 0.800i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.660922474 - 1.830962795i\)
\(L(\frac12)\) \(\approx\) \(3.660922474 - 1.830962795i\)
\(L(1)\) \(\approx\) \(2.196686544 - 0.4870232234i\)
\(L(1)\) \(\approx\) \(2.196686544 - 0.4870232234i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.966 - 0.255i)T \)
5 \( 1 + (0.665 + 0.746i)T \)
11 \( 1 + (0.802 - 0.596i)T \)
13 \( 1 + (-0.986 + 0.162i)T \)
17 \( 1 + (-0.00679 - 0.999i)T \)
19 \( 1 + (0.786 - 0.618i)T \)
29 \( 1 + (0.862 - 0.505i)T \)
31 \( 1 + (0.888 - 0.458i)T \)
37 \( 1 + (-0.694 - 0.719i)T \)
41 \( 1 + (-0.979 + 0.202i)T \)
43 \( 1 + (-0.714 - 0.699i)T \)
47 \( 1 + (0.826 + 0.563i)T \)
53 \( 1 + (0.612 + 0.790i)T \)
59 \( 1 + (-0.833 - 0.552i)T \)
61 \( 1 + (-0.810 + 0.585i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (-0.182 - 0.983i)T \)
73 \( 1 + (0.675 + 0.737i)T \)
79 \( 1 + (0.235 + 0.971i)T \)
83 \( 1 + (-0.768 + 0.639i)T \)
89 \( 1 + (0.601 + 0.798i)T \)
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

−19.1430623009177728178070907213, −17.91640422974819647515498697328, −17.27931231565041352733607034146, −16.87772755848093076532968016784, −16.166933998960767989376510129107, −15.30358246680221627152640726131, −14.71150374826516298540559013479, −14.013263879061158848113081564, −13.47452507829857923043267431434, −12.556597168949399980551692835263, −12.19049532877061558208012452828, −11.6470242777076510426519866282, −10.232414882104931765584097170364, −10.0873563400210131481901724977, −8.88713012772200782057295636402, −8.2521926882238153309879759407, −7.33930559815346625960848685463, −6.58716089723315172311845701793, −5.95077625864160767686127406173, −5.04160694916126760706647667410, −4.676449837132495762103346806847, −3.7336782310170184275860743870, −2.87474400594085260489482682540, −1.86027335982530410875758774035, −1.31479665211977586672318956280, 0.83154280060458684103309549426, 1.89024515844596461003358430524, 2.73110541199766845152783335303, 3.16395436045269994293858248586, 4.20377276310102855739370206614, 5.01932045805960904173221885697, 5.671832607600181387765644440753, 6.55033348488964793773796461762, 6.96981609365290676990262508656, 7.76016561531695664541543301482, 9.11320990095570459118773047900, 9.68968290989701791106120565728, 10.398792438021979852907515223969, 11.16183195727076442602334999245, 11.85404726440491424539346325842, 12.27244561360110114900444160606, 13.54399101740480675458941942768, 13.82724411450187025303933230712, 14.27101334091676486184847173974, 15.21545800711246096771488836995, 15.63272656090993799363298960793, 16.71075587101816015255703698059, 17.17265443005385887665727394598, 18.14257185717141753590212301691, 18.83712179223869698270414651363

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