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Properties

Label 1-1441-1441.890-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.979 - 0.203i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.527 − 0.849i)2-s + (−0.681 − 0.732i)3-s + (−0.443 + 0.896i)4-s + (−0.906 + 0.421i)5-s + (−0.262 + 0.964i)6-s + (0.998 + 0.0483i)7-s + (0.995 − 0.0965i)8-s + (−0.0724 + 0.997i)9-s + (0.836 + 0.548i)10-s + (0.958 − 0.285i)12-s + (−0.836 + 0.548i)13-s + (−0.485 − 0.873i)14-s + (0.926 + 0.377i)15-s + (−0.607 − 0.794i)16-s + (−0.943 + 0.331i)17-s + (0.885 − 0.464i)18-s + ⋯
L(s)  = 1  + (−0.527 − 0.849i)2-s + (−0.681 − 0.732i)3-s + (−0.443 + 0.896i)4-s + (−0.906 + 0.421i)5-s + (−0.262 + 0.964i)6-s + (0.998 + 0.0483i)7-s + (0.995 − 0.0965i)8-s + (−0.0724 + 0.997i)9-s + (0.836 + 0.548i)10-s + (0.958 − 0.285i)12-s + (−0.836 + 0.548i)13-s + (−0.485 − 0.873i)14-s + (0.926 + 0.377i)15-s + (−0.607 − 0.794i)16-s + (−0.943 + 0.331i)17-s + (0.885 − 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.979 - 0.203i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (890, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.979 - 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5453324842 - 0.05598658442i\)
\(L(\frac12)\) \(\approx\) \(0.5453324842 - 0.05598658442i\)
\(L(1)\) \(\approx\) \(0.5091417108 - 0.2023089149i\)
\(L(1)\) \(\approx\) \(0.5091417108 - 0.2023089149i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.527 - 0.849i)T \)
3 \( 1 + (-0.681 - 0.732i)T \)
5 \( 1 + (-0.906 + 0.421i)T \)
7 \( 1 + (0.998 + 0.0483i)T \)
13 \( 1 + (-0.836 + 0.548i)T \)
17 \( 1 + (-0.943 + 0.331i)T \)
19 \( 1 + (0.568 + 0.822i)T \)
23 \( 1 + (-0.215 - 0.976i)T \)
29 \( 1 + (-0.0724 - 0.997i)T \)
31 \( 1 + (-0.399 - 0.916i)T \)
37 \( 1 + (-0.715 - 0.698i)T \)
41 \( 1 + (0.607 - 0.794i)T \)
43 \( 1 + (-0.981 - 0.192i)T \)
47 \( 1 + (0.970 - 0.239i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.0241 + 0.999i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.981 + 0.192i)T \)
71 \( 1 + (0.748 - 0.663i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (0.120 + 0.992i)T \)
83 \( 1 + (-0.168 + 0.985i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.779 - 0.626i)T \)
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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

−20.44225815664608246650788298132, −20.0936058436906166718226844368, −19.23931956461670686971010114453, −17.92134628046260519998120008704, −17.80740507751362916240149675662, −16.94962903715171590262551298860, −16.16982245295897101164116444404, −15.56410103976632282077508928311, −15.04109213846893905800838586854, −14.31119007226896980374111381443, −13.19991796610948948031183986469, −12.09724793149383377793244258219, −11.324476338867085332972628960578, −10.80906731004019372777093220599, −9.81361939024291678038637544959, −9.007735844095382162914360086932, −8.34474896081681259032047070086, −7.3857662084829647813550339066, −6.842223995919389200246845490623, −5.4665320927652456331535794396, −4.96106186652878048455449590784, −4.47596292139840291793947551655, −3.27920906536582827510693184655, −1.542369707982677560131830314197, −0.425327710959779026584050612841, 0.74990702780060400264126644703, 1.99210396598388828013648156349, 2.47781888533746849689986033758, 4.01451480238239405833741130754, 4.49338951992057686621892158966, 5.63472940312237336882049588661, 6.93056539519256458429340447256, 7.51301138361677817036549334570, 8.15408452243128061046891151078, 8.95448404209344492647373590801, 10.28943595330579715812424591801, 10.80940031938446172241625504496, 11.68281473113884598023134441554, 11.92041304931593870215019229790, 12.69863948862894443780630684154, 13.74862784746634167727405375243, 14.447927556569745825183863414571, 15.4986367635999081075216580946, 16.56828210037906479905928358457, 17.10003462323354303148246520632, 17.938854107234706300680378345953, 18.498174976363845047554593110027, 19.08724424369481119214948185650, 19.79718006723758369720844421033, 20.52016043381698261387262386286

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