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Properties

Label 1-1441-1441.625-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: $\chi_{1441} (625, \cdot )$
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

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

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