↑

Properties

Label 1-1441-1441.108-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.971 - 0.238i$
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  + (−0.168 + 0.985i)2-s + (−0.970 − 0.239i)3-s + (−0.943 − 0.331i)4-s + (−0.970 − 0.239i)5-s + (0.399 − 0.916i)6-s + (0.399 + 0.916i)7-s + (0.485 − 0.873i)8-s + (0.885 + 0.464i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (0.215 − 0.976i)13-s + (−0.970 + 0.239i)14-s + (0.885 + 0.464i)15-s + (0.779 + 0.626i)16-s + (−0.998 − 0.0483i)17-s + (−0.607 + 0.794i)18-s + ⋯
L(s)  = 1  + (−0.168 + 0.985i)2-s + (−0.970 − 0.239i)3-s + (−0.943 − 0.331i)4-s + (−0.970 − 0.239i)5-s + (0.399 − 0.916i)6-s + (0.399 + 0.916i)7-s + (0.485 − 0.873i)8-s + (0.885 + 0.464i)9-s + (0.399 − 0.916i)10-s + (0.836 + 0.548i)12-s + (0.215 − 0.976i)13-s + (−0.970 + 0.239i)14-s + (0.885 + 0.464i)15-s + (0.779 + 0.626i)16-s + (−0.998 − 0.0483i)17-s + (−0.607 + 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.971 - 0.238i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (108, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.971 - 0.238i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02315179219 + 0.1910282479i\)
\(L(\frac12)\) \(\approx\) \(0.02315179219 + 0.1910282479i\)
\(L(1)\) \(\approx\) \(0.4532084984 + 0.1994187882i\)
\(L(1)\) \(\approx\) \(0.4532084984 + 0.1994187882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.168 + 0.985i)T \)
3 \( 1 + (-0.970 - 0.239i)T \)
5 \( 1 + (-0.970 - 0.239i)T \)
7 \( 1 + (0.399 + 0.916i)T \)
13 \( 1 + (0.215 - 0.976i)T \)
17 \( 1 + (-0.998 - 0.0483i)T \)
19 \( 1 + (-0.998 + 0.0483i)T \)
23 \( 1 + (-0.906 - 0.421i)T \)
29 \( 1 + (-0.168 - 0.985i)T \)
31 \( 1 + (0.120 + 0.992i)T \)
37 \( 1 + (0.958 - 0.285i)T \)
41 \( 1 + (0.836 + 0.548i)T \)
43 \( 1 + (0.926 - 0.377i)T \)
47 \( 1 + (0.715 - 0.698i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.262 - 0.964i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.926 + 0.377i)T \)
71 \( 1 + (-0.681 - 0.732i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.644 + 0.764i)T \)
83 \( 1 + (0.568 - 0.822i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (-0.748 + 0.663i)T \)
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

−20.34320612411235599529387638489, −19.56226304301620217207778268769, −18.90790672138720482937561637414, −18.07747774239240205374809475419, −17.45248019871709753055011719521, −16.66608418172122533857183121067, −16.04589665632598410097006962133, −14.97944845040575972155499495747, −14.14636442941336933124049801298, −13.19154791370438537308043446515, −12.45430595227022937054128989855, −11.59497574830202367515325843276, −11.08334424352796119665322773507, −10.70459011665101795525345847996, −9.72083318873547841658830184827, −8.822050804553430433538398044117, −7.82699903907305383783168105961, −7.06497889524261716398950379494, −6.10028646557563697277068501246, −4.74417048063027581008360589868, −4.17472569125814976451008249519, −3.8163844624889590948743071021, −2.30976882353870566365548435956, −1.2263688423864934314996491233, −0.12377297077791384514045342107, 0.96546172211058471541893560900, 2.38245815250483743744466722835, 4.0661618048132436079487105773, 4.56052954457085315744010580314, 5.53910780439473876680068905407, 6.10538528807434101905589781530, 6.99243768736357708337041614014, 7.934036784274573975206914019072, 8.37530072768229899634421877668, 9.303407209964824960994250716167, 10.49764808218954474675932646044, 11.10059075562248158049908505075, 12.130726075965687785035797792261, 12.65230299762109383256255139313, 13.427658364901027231971541881814, 14.66323685036438121482616046109, 15.37427393676121234364280120025, 15.8135907644529925479966491604, 16.49708653588038780618984978549, 17.48267845664101921597241902719, 17.89351482553818722041613415397, 18.6821910624893591189744059179, 19.29017499982602940574175955534, 20.23007780992468639362898124499, 21.459424790786825613402801901595

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