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Properties

Label 1-1441-1441.1358-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.705 - 0.708i$
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  + (0.995 − 0.0965i)2-s + (−0.262 + 0.964i)3-s + (0.981 − 0.192i)4-s + (−0.998 + 0.0483i)5-s + (−0.168 + 0.985i)6-s + (0.885 − 0.464i)7-s + (0.958 − 0.285i)8-s + (−0.861 − 0.506i)9-s + (−0.989 + 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.885 − 0.464i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (0.926 − 0.377i)16-s + (−0.0724 − 0.997i)17-s + (−0.906 − 0.421i)18-s + ⋯
L(s)  = 1  + (0.995 − 0.0965i)2-s + (−0.262 + 0.964i)3-s + (0.981 − 0.192i)4-s + (−0.998 + 0.0483i)5-s + (−0.168 + 0.985i)6-s + (0.885 − 0.464i)7-s + (0.958 − 0.285i)8-s + (−0.861 − 0.506i)9-s + (−0.989 + 0.144i)10-s + (−0.0724 + 0.997i)12-s + (0.885 − 0.464i)13-s + (0.836 − 0.548i)14-s + (0.215 − 0.976i)15-s + (0.926 − 0.377i)16-s + (−0.0724 − 0.997i)17-s + (−0.906 − 0.421i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.705 - 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.705 - 0.708i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1358, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.705 - 0.708i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.092598162 - 0.8698013213i\)
\(L(\frac12)\) \(\approx\) \(2.092598162 - 0.8698013213i\)
\(L(1)\) \(\approx\) \(1.610208535 - 0.04618774052i\)
\(L(1)\) \(\approx\) \(1.610208535 - 0.04618774052i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.995 - 0.0965i)T \)
3 \( 1 + (-0.262 + 0.964i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (-0.0724 - 0.997i)T \)
19 \( 1 + (-0.527 - 0.849i)T \)
23 \( 1 + (-0.943 + 0.331i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (-0.607 + 0.794i)T \)
37 \( 1 + (0.120 - 0.992i)T \)
41 \( 1 + (0.644 - 0.764i)T \)
43 \( 1 + (-0.998 + 0.0483i)T \)
47 \( 1 + (-0.861 - 0.506i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.0724 - 0.997i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (-0.998 - 0.0483i)T \)
71 \( 1 + (0.0241 - 0.999i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.836 + 0.548i)T \)
83 \( 1 + (0.981 + 0.192i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.715 + 0.698i)T \)
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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

−20.73625637898736126643450442332, −20.22323635934522049898382065643, −19.29955689709040687742113163228, −18.675404832046563201822197440467, −17.93393432824894248572472449659, −16.75038722214657383696614663366, −16.3951171585131124971965083458, −15.23553241846190807883723586163, −14.72126131649505177558348520618, −14.01780214787268612828059682693, −13.01253751398569656369289899425, −12.49776370462772441065351387849, −11.69929949738816737179805358112, −11.2717190767606998367887768724, −10.55293881753786160757430730960, −8.71398879680286190153841922464, −8.030027731128575784994218906835, −7.612009335393786758520914305926, −6.39410657878807577610114777946, −6.00616305830659300118622531174, −4.91388935554869595053183011494, −4.107276096405039446825234466956, −3.24929722794402956356774967134, −1.97651787484964589729663829308, −1.442516221443681726018653834983, 0.621834899183355855273445317085, 2.1187850732649696114861055842, 3.38257511268850673803131386850, 3.83305903557594195483094672885, 4.69008777920283273202992640875, 5.21973772947207070412675012347, 6.25349474713666354830016737438, 7.26664101923230281883731275946, 8.026828012422415790223036265253, 8.96969100386245830050312007276, 10.19671497124553043965609427901, 10.99838865714369701372958539374, 11.32031906635329190681244319895, 11.99097135205854332988086056276, 13.04896523958567610856242116103, 13.96053495056565890672921405403, 14.649030086496497477072002101239, 15.31922030107295144395217594203, 15.95076240863117023703029985555, 16.44574478507329282484016507666, 17.48924596870464729966125305887, 18.31297433291721630870373796587, 19.64446010731041879829402310532, 20.06398354512928802513447572841, 20.81976188113139870952956923877

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