↑

Properties

Label 1-1840-1840.1603-r0-0-0
Degree $1$
Conductor $1840$
Sign $0.0298 + 0.999i$
Analytic cond. $8.54492$
Root an. cond. $8.54492$
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  + (0.415 − 0.909i)3-s + (−0.540 − 0.841i)7-s + (−0.654 − 0.755i)9-s + (−0.989 − 0.142i)11-s + (−0.841 − 0.540i)13-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.989 + 0.142i)21-s + (−0.959 + 0.281i)27-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.540 + 0.841i)33-s + (0.654 + 0.755i)37-s + (−0.841 + 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)3-s + (−0.540 − 0.841i)7-s + (−0.654 − 0.755i)9-s + (−0.989 − 0.142i)11-s + (−0.841 − 0.540i)13-s + (−0.281 + 0.959i)17-s + (−0.281 − 0.959i)19-s + (−0.989 + 0.142i)21-s + (−0.959 + 0.281i)27-s + (−0.281 + 0.959i)29-s + (−0.415 − 0.909i)31-s + (−0.540 + 0.841i)33-s + (0.654 + 0.755i)37-s + (−0.841 + 0.540i)39-s + (0.654 − 0.755i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0298 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0298 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.0298 + 0.999i$
Analytic conductor: \(8.54492\)
Root analytic conductor: \(8.54492\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1603, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1840,\ (0:\ ),\ 0.0298 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01859865154 + 0.01805156884i\)
\(L(\frac12)\) \(\approx\) \(0.01859865154 + 0.01805156884i\)
\(L(1)\) \(\approx\) \(0.7255714556 - 0.3475437471i\)
\(L(1)\) \(\approx\) \(0.7255714556 - 0.3475437471i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 \)
good3 \( 1 + (0.415 - 0.909i)T \)
7 \( 1 + (-0.540 - 0.841i)T \)
11 \( 1 + (-0.989 - 0.142i)T \)
13 \( 1 + (-0.841 - 0.540i)T \)
17 \( 1 + (-0.281 + 0.959i)T \)
19 \( 1 + (-0.281 - 0.959i)T \)
29 \( 1 + (-0.281 + 0.959i)T \)
31 \( 1 + (-0.415 - 0.909i)T \)
37 \( 1 + (0.654 + 0.755i)T \)
41 \( 1 + (0.654 - 0.755i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 - iT \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (0.540 - 0.841i)T \)
61 \( 1 + (-0.909 + 0.415i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
71 \( 1 + (-0.142 - 0.989i)T \)
73 \( 1 + (0.281 + 0.959i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.415 - 0.909i)T \)
97 \( 1 + (0.755 + 0.654i)T \)
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

−19.988599946611342726454421843417, −19.31819269513699952619876866062, −18.5591343960199742457933262959, −17.883787235566223047388740899030, −16.599750410635400231985311385564, −16.3887968761522451464169060141, −15.40840445201511405792825825178, −15.03795915952011247844812044299, −14.13473530859889707883392733551, −13.40531345921847463559601700957, −12.46344169140977615705664278894, −11.78505249728014765939778201920, −10.84567372832322394014654897269, −10.020739902644926729406565631127, −9.49095079119953438891165959562, −8.78527946161422365521989171757, −7.93207968727589966508248199417, −7.10235759986434699777034142391, −5.91244250820405335226828143886, −5.250877999750364647669978915628, −4.49088070851996156888580111885, −3.5228329338093876584937062114, −2.58537907424488749303466167332, −2.13065808728008862349078634020, −0.008864818276297215934539513684, 1.04814582745921805473122812954, 2.27258619079234840217539964635, 2.917551550444841333373113507909, 3.837503257054762319266839026818, 4.90932095920356445159496975550, 5.92123825914449575652792466634, 6.70559274130645173550726829344, 7.48115639619885341348833474305, 7.97415289351819872741492939252, 8.92279822713521513970083534387, 9.78995396248546780132766382047, 10.618747280083210396536414211089, 11.321339713736047506949870186410, 12.4827749098829558718846956499, 13.07035470330402681594803537484, 13.32534985513694650935007347549, 14.42369419646688644931291071743, 15.0073322209561782135585894006, 15.87198072277875388951495203242, 16.86095890944451548921682352773, 17.44410540624414850191660281854, 18.14037263586118055133283912716, 18.96671272263422942895531135463, 19.65393466562907991051253138119, 20.07654318728295561471055312813

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