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Properties

Label 1-3381-3381.1214-r0-0-0
Degree $1$
Conductor $3381$
Sign $0.827 + 0.561i$
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  + (−0.601 + 0.798i)2-s + (−0.275 − 0.961i)4-s + (0.511 + 0.859i)5-s + (0.933 + 0.359i)8-s + (−0.994 − 0.108i)10-s + (−0.999 − 0.0271i)11-s + (−0.917 − 0.396i)13-s + (−0.848 + 0.529i)16-s + (−0.694 − 0.719i)17-s + (0.995 + 0.0950i)19-s + (0.685 − 0.728i)20-s + (0.623 − 0.781i)22-s + (−0.476 + 0.879i)25-s + (0.869 − 0.494i)26-s + (−0.970 + 0.242i)29-s + ⋯
L(s)  = 1  + (−0.601 + 0.798i)2-s + (−0.275 − 0.961i)4-s + (0.511 + 0.859i)5-s + (0.933 + 0.359i)8-s + (−0.994 − 0.108i)10-s + (−0.999 − 0.0271i)11-s + (−0.917 − 0.396i)13-s + (−0.848 + 0.529i)16-s + (−0.694 − 0.719i)17-s + (0.995 + 0.0950i)19-s + (0.685 − 0.728i)20-s + (0.623 − 0.781i)22-s + (−0.476 + 0.879i)25-s + (0.869 − 0.494i)26-s + (−0.970 + 0.242i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3381\)    =    \(3 \cdot 7^{2} \cdot 23\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(15.7012\)
Root analytic conductor: \(15.7012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3381} (1214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3381,\ (0:\ ),\ 0.827 + 0.561i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8343532629 + 0.2563653675i\)
\(L(\frac12)\) \(\approx\) \(0.8343532629 + 0.2563653675i\)
\(L(1)\) \(\approx\) \(0.6677249648 + 0.2685192190i\)
\(L(1)\) \(\approx\) \(0.6677249648 + 0.2685192190i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.601 + 0.798i)T \)
5 \( 1 + (0.511 + 0.859i)T \)
11 \( 1 + (-0.999 - 0.0271i)T \)
13 \( 1 + (-0.917 - 0.396i)T \)
17 \( 1 + (-0.694 - 0.719i)T \)
19 \( 1 + (0.995 + 0.0950i)T \)
29 \( 1 + (-0.970 + 0.242i)T \)
31 \( 1 + (-0.928 + 0.371i)T \)
37 \( 1 + (0.906 - 0.421i)T \)
41 \( 1 + (0.488 - 0.872i)T \)
43 \( 1 + (-0.933 + 0.359i)T \)
47 \( 1 + (0.0747 - 0.997i)T \)
53 \( 1 + (-0.760 + 0.649i)T \)
59 \( 1 + (0.994 + 0.108i)T \)
61 \( 1 + (0.00679 + 0.999i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.947 + 0.320i)T \)
73 \( 1 + (0.876 - 0.482i)T \)
79 \( 1 + (0.981 - 0.189i)T \)
83 \( 1 + (0.986 + 0.162i)T \)
89 \( 1 + (-0.675 - 0.737i)T \)
97 \( 1 + (0.959 + 0.281i)T \)
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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

−18.76722459715261434665007913391, −17.999951255548674049024370153207, −17.56186394186419018336358861146, −16.73046928982534947104783674641, −16.34412534505557758807116777531, −15.46401674203730904123234445396, −14.49454100716790606755703616326, −13.55400485101150404721549614072, −12.97799683956960777077428107733, −12.61209616971732628247647328914, −11.64245911756437748992142752514, −11.099613724878412126913137726515, −10.150915775965806454775869518923, −9.58633614483885911256373365032, −9.11961419170293319818724747606, −8.12832042928845998856556764919, −7.72089436277789584812869026264, −6.73725921828141265813631704175, −5.62153091860158302110775964844, −4.88945238194759202562371525360, −4.23177725832398670532634253801, −3.20147020769171569074735780940, −2.26236206747464053573694156295, −1.77215127881931752632635469029, −0.65442105232718290712356477205, 0.46733824076835017091854388731, 1.86573571721508456577285282247, 2.5155547221288329182930761116, 3.44881417154489610754616982252, 4.76715152680248374275198577799, 5.39338395387662178120513607671, 5.949860001110040031664582020267, 7.06652995175797159322151574185, 7.314170328778957256809204368128, 8.03582290798161909807410693797, 9.16282007314544901741352394641, 9.591902258803705224372899532054, 10.37972857339561660942263794761, 10.88114452116258038068613083049, 11.68449771545825305275863066037, 12.89546961108760467080852830248, 13.51450670986026147222384240740, 14.19741235747871713289139829100, 14.881247031617403906795335124367, 15.389022171835492945796398676612, 16.179140001072502484790784540067, 16.792352288274925852998844770344, 17.73205512397136010786575116676, 18.12337951776522489106049820535, 18.52144499992865620543633516854

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