↑

Properties

Label 1-1441-1441.447-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.876 + 0.481i$
Analytic cond. $6.69197$
Root an. cond. $6.69197$
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  + (−0.681 + 0.732i)2-s + (0.215 − 0.976i)3-s + (−0.0724 − 0.997i)4-s + (−0.861 − 0.506i)5-s + (0.568 + 0.822i)6-s + (0.607 + 0.794i)7-s + (0.779 + 0.626i)8-s + (−0.906 − 0.421i)9-s + (0.958 − 0.285i)10-s + (−0.989 − 0.144i)12-s + (−0.0241 − 0.999i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (−0.989 + 0.144i)16-s + (0.885 + 0.464i)17-s + (0.926 − 0.377i)18-s + ⋯
L(s)  = 1  + (−0.681 + 0.732i)2-s + (0.215 − 0.976i)3-s + (−0.0724 − 0.997i)4-s + (−0.861 − 0.506i)5-s + (0.568 + 0.822i)6-s + (0.607 + 0.794i)7-s + (0.779 + 0.626i)8-s + (−0.906 − 0.421i)9-s + (0.958 − 0.285i)10-s + (−0.989 − 0.144i)12-s + (−0.0241 − 0.999i)13-s + (−0.995 − 0.0965i)14-s + (−0.681 + 0.732i)15-s + (−0.989 + 0.144i)16-s + (0.885 + 0.464i)17-s + (0.926 − 0.377i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.876 + 0.481i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.876 + 0.481i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.003872798558 + 0.01507742657i\)
\(L(\frac12)\) \(\approx\) \(0.003872798558 + 0.01507742657i\)
\(L(1)\) \(\approx\) \(0.5827062637 - 0.05053693834i\)
\(L(1)\) \(\approx\) \(0.5827062637 - 0.05053693834i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.681 + 0.732i)T \)
3 \( 1 + (0.215 - 0.976i)T \)
5 \( 1 + (-0.861 - 0.506i)T \)
7 \( 1 + (0.607 + 0.794i)T \)
13 \( 1 + (-0.0241 - 0.999i)T \)
17 \( 1 + (0.885 + 0.464i)T \)
19 \( 1 + (-0.989 - 0.144i)T \)
23 \( 1 + (-0.779 + 0.626i)T \)
29 \( 1 + (0.981 + 0.192i)T \)
31 \( 1 + (-0.779 - 0.626i)T \)
37 \( 1 + (0.527 - 0.849i)T \)
41 \( 1 + (-0.885 + 0.464i)T \)
43 \( 1 + (-0.995 + 0.0965i)T \)
47 \( 1 + (0.681 - 0.732i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.168 + 0.985i)T \)
61 \( 1 - T \)
67 \( 1 + (-0.995 - 0.0965i)T \)
71 \( 1 + (-0.779 - 0.626i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (-0.861 - 0.506i)T \)
83 \( 1 + (-0.527 - 0.849i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.607 + 0.794i)T \)
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

−20.331601140976999038200266685028, −19.80504421334634221624171661664, −18.99513686773572040834469506878, −18.377460091758375335630108785061, −17.30347806305357890743782931991, −16.628873754605298000656136748992, −16.14851547084843741694961751285, −15.16149333224765428548455179789, −14.26001379448420748374083753890, −13.82359012507427933970461638515, −12.36585544112701133049603113601, −11.71126896878412470882526732025, −11.01752781583236873617127247457, −10.41690043159497573154888369201, −9.82255562005336701342274712208, −8.721110719426012238621267788982, −8.169238816994008817464542356435, −7.40165753375476821399111609671, −6.4608212227922221310187188977, −4.74260602306900887927143388440, −4.27698987519105462738346129370, −3.52185746643711478596947442246, −2.69524303837542804894398597618, −1.5392917039669276882066902668, −0.00775710912706589901815033005, 1.20482154831148489408117703482, 2.01508885408745605269836145827, 3.24138450502063002283161697520, 4.56961409665102652794435317998, 5.571544362607410367986829107640, 6.07218249990356268423713555026, 7.2876547649149856143025419839, 7.87167935608990466360079057626, 8.41081365901556434730365444948, 8.945321846128478055494538990349, 10.130875868147111155939616878820, 11.144639171691272809655051056692, 11.907271065952828403909815128687, 12.61342331247416240241380568035, 13.436945652121250155043871897892, 14.57314926412689751035884672305, 14.98212789153452345950165315830, 15.68682880814549662742981004914, 16.67880261165029041377540138093, 17.334108666582925680040565634746, 18.15006078003026017623660030081, 18.608063254280606371443468715852, 19.50471951813454473937690905224, 19.876990319121032834175401252063, 20.72392093806928623979348673891

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