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Properties

Label 1-1441-1441.65-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.120 + 0.992i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.998 − 0.0483i)2-s + (−0.943 + 0.331i)3-s + (0.995 − 0.0965i)4-s + (−0.607 − 0.794i)5-s + (−0.926 + 0.377i)6-s + (0.0724 − 0.997i)7-s + (0.989 − 0.144i)8-s + (0.779 − 0.626i)9-s + (−0.644 − 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.644 + 0.764i)13-s + (0.0241 − 0.999i)14-s + (0.836 + 0.548i)15-s + (0.981 − 0.192i)16-s + (−0.485 − 0.873i)17-s + (0.748 − 0.663i)18-s + ⋯
L(s)  = 1  + (0.998 − 0.0483i)2-s + (−0.943 + 0.331i)3-s + (0.995 − 0.0965i)4-s + (−0.607 − 0.794i)5-s + (−0.926 + 0.377i)6-s + (0.0724 − 0.997i)7-s + (0.989 − 0.144i)8-s + (0.779 − 0.626i)9-s + (−0.644 − 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.644 + 0.764i)13-s + (0.0241 − 0.999i)14-s + (0.836 + 0.548i)15-s + (0.981 − 0.192i)16-s + (−0.485 − 0.873i)17-s + (0.748 − 0.663i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.120 + 0.992i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.120 + 0.992i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3509442509 + 0.3962345909i\)
\(L(\frac12)\) \(\approx\) \(0.3509442509 + 0.3962345909i\)
\(L(1)\) \(\approx\) \(1.104141148 - 0.2218975342i\)
\(L(1)\) \(\approx\) \(1.104141148 - 0.2218975342i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.998 - 0.0483i)T \)
3 \( 1 + (-0.943 + 0.331i)T \)
5 \( 1 + (-0.607 - 0.794i)T \)
7 \( 1 + (0.0724 - 0.997i)T \)
13 \( 1 + (-0.644 + 0.764i)T \)
17 \( 1 + (-0.485 - 0.873i)T \)
19 \( 1 + (-0.120 - 0.992i)T \)
23 \( 1 + (-0.443 + 0.896i)T \)
29 \( 1 + (-0.779 - 0.626i)T \)
31 \( 1 + (-0.168 + 0.985i)T \)
37 \( 1 + (0.399 + 0.916i)T \)
41 \( 1 + (-0.981 - 0.192i)T \)
43 \( 1 + (-0.958 - 0.285i)T \)
47 \( 1 + (-0.354 - 0.935i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (-0.681 + 0.732i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.958 - 0.285i)T \)
71 \( 1 + (0.885 + 0.464i)T \)
73 \( 1 + (0.809 + 0.587i)T \)
79 \( 1 + (-0.568 + 0.822i)T \)
83 \( 1 + (0.861 - 0.506i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.527 - 0.849i)T \)
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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

−20.39734878710581674713879840208, −19.57723660996168335116956753193, −18.753445726085842207600351629225, −18.22788996196882637641813305968, −17.207579402801320276157662727568, −16.41630683528314544048429670611, −15.65366688142870148468445350322, −14.93170208183567609017751894420, −14.53214625997217500142344068053, −13.24321346108037756874487304651, −12.47368674972152475902996897761, −12.14822552293795562004639364574, −11.23032152093689976990107550524, −10.7190040043000082183561929049, −9.88253465495780736222945479276, −8.19249580909896400971721465452, −7.672294471894444954010785242585, −6.651426694844683265878423398617, −6.07398327430806177427040657631, −5.39009070369048364938135952873, −4.4508018170692546942124776293, −3.56851502684621843224374624834, −2.490711626346137979731035859584, −1.771764410247607822320309081634, −0.08770939598016149953660102043, 0.89706698033528970380589969822, 1.94853067129982473897766053910, 3.50000430930790646180923197613, 4.11696114081564034451029976784, 4.96169882103394792417953653591, 5.21025436891761278214461898688, 6.76304650174409294576454639554, 6.95541224408917950916849410861, 8.01084100834394657893704777486, 9.3802066801325690418446217840, 10.09986889102153948455132778542, 11.197418468260637379977137296804, 11.55621565467648625550490880211, 12.19045406911457119085198015703, 13.22107214526916706097338720991, 13.60437980628976057079064112740, 14.76997240525576901499833500579, 15.5878577960153355180101799051, 16.10467277956650525892817667555, 16.94494428987572481817243846989, 17.21017226183563967852268170882, 18.50285733830966839551872638843, 19.69954876375727058405386290314, 20.04432434374920447122469636156, 20.87930675058076533855035156491

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