↑

Properties

Label 1-6009-6009.1259-r0-0-0
Degree $1$
Conductor $6009$
Sign $0.993 + 0.114i$
Analytic cond. $27.9056$
Root an. cond. $27.9056$
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  + (0.375 + 0.927i)2-s + (−0.718 + 0.695i)4-s + (0.925 − 0.379i)5-s + (0.735 − 0.677i)7-s + (−0.914 − 0.405i)8-s + (0.698 + 0.715i)10-s + (0.975 − 0.217i)11-s + (−0.975 + 0.220i)13-s + (0.903 + 0.428i)14-s + (0.0329 − 0.999i)16-s + (0.978 − 0.205i)17-s + (−0.440 − 0.897i)19-s + (−0.401 + 0.916i)20-s + (0.568 + 0.822i)22-s + (−0.875 + 0.484i)23-s + ⋯
L(s)  = 1  + (0.375 + 0.927i)2-s + (−0.718 + 0.695i)4-s + (0.925 − 0.379i)5-s + (0.735 − 0.677i)7-s + (−0.914 − 0.405i)8-s + (0.698 + 0.715i)10-s + (0.975 − 0.217i)11-s + (−0.975 + 0.220i)13-s + (0.903 + 0.428i)14-s + (0.0329 − 0.999i)16-s + (0.978 − 0.205i)17-s + (−0.440 − 0.897i)19-s + (−0.401 + 0.916i)20-s + (0.568 + 0.822i)22-s + (−0.875 + 0.484i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6009 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.114i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6009\)    =    \(3 \cdot 2003\)
Sign: $0.993 + 0.114i$
Analytic conductor: \(27.9056\)
Root analytic conductor: \(27.9056\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6009} (1259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6009,\ (0:\ ),\ 0.993 + 0.114i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.738525404 + 0.1567247681i\)
\(L(\frac12)\) \(\approx\) \(2.738525404 + 0.1567247681i\)
\(L(1)\) \(\approx\) \(1.480488674 + 0.4112145647i\)
\(L(1)\) \(\approx\) \(1.480488674 + 0.4112145647i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
2003 \( 1 \)
good2 \( 1 + (0.375 + 0.927i)T \)
5 \( 1 + (0.925 - 0.379i)T \)
7 \( 1 + (0.735 - 0.677i)T \)
11 \( 1 + (0.975 - 0.217i)T \)
13 \( 1 + (-0.975 + 0.220i)T \)
17 \( 1 + (0.978 - 0.205i)T \)
19 \( 1 + (-0.440 - 0.897i)T \)
23 \( 1 + (-0.875 + 0.484i)T \)
29 \( 1 + (0.310 + 0.950i)T \)
31 \( 1 + (0.961 + 0.275i)T \)
37 \( 1 + (-0.967 - 0.254i)T \)
41 \( 1 + (0.999 - 0.0313i)T \)
43 \( 1 + (0.111 - 0.993i)T \)
47 \( 1 + (-0.929 - 0.367i)T \)
53 \( 1 + (-0.813 - 0.582i)T \)
59 \( 1 + (0.856 - 0.516i)T \)
61 \( 1 + (0.836 + 0.548i)T \)
67 \( 1 + (0.395 + 0.918i)T \)
71 \( 1 + (-0.0109 - 0.999i)T \)
73 \( 1 + (0.0893 + 0.996i)T \)
79 \( 1 + (0.742 - 0.670i)T \)
83 \( 1 + (0.477 - 0.878i)T \)
89 \( 1 + (-0.885 + 0.464i)T \)
97 \( 1 + (-0.280 - 0.959i)T \)
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

−17.710901898073476213150166582036, −17.362342613254287957392505153612, −16.61290920430560404848861625026, −15.39523507879036679238584519349, −14.75541835223233014826109085552, −14.26423952845660643326015770856, −14.03479065878033784622620441766, −12.92673795195020468702776745492, −12.27036021496151788552504888825, −11.95539158682280609424718722280, −11.14288944027415143102711872855, −10.37714733700267617074793926735, −9.72345564480181620453291046110, −9.48632082449934878866594226774, −8.37919550721909212033141763366, −7.900072975995408581438045765024, −6.58548887994146716336448713042, −6.04328322841253327726418270300, −5.428227443309233158534987292545, −4.6718798724829892461634625932, −3.99958065184726395253217511904, −3.01182078416635726515458868447, −2.34454168937228165178771216299, −1.77430517707827456352426850897, −1.05593656753454046792730453954, 0.63946106748037454233816083179, 1.53381047087951548662532444573, 2.46773172288328661409223528754, 3.5051299289949298691635506630, 4.23020970953658768881814481474, 5.0158803782978842720407276743, 5.327620549038453683144314892702, 6.325579113910083470279873083992, 6.88239178335856941657083542686, 7.495583661812047030089657619287, 8.38184777461253869323654293407, 8.873390268872248952244785117927, 9.7196874658251759368348230680, 10.13863948658293981755915908793, 11.23648358168763341007866863836, 12.05173159790361916108132038504, 12.50426120872205892499608450720, 13.45051944013099478397156770072, 13.92227421784439767023215441204, 14.46763883240081574632757397887, 14.782547250016297340451216700684, 15.96743179618036617868514758665, 16.42167074338018440522019510882, 17.18621182609947329418637179112, 17.5092543240508573970914339825

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