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Properties

Label 1-1441-1441.357-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.997 - 0.0722i$
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  + (0.989 + 0.144i)2-s + (0.926 + 0.377i)3-s + (0.958 + 0.285i)4-s + (−0.0724 + 0.997i)5-s + (0.861 + 0.506i)6-s + (−0.748 − 0.663i)7-s + (0.906 + 0.421i)8-s + (0.715 + 0.698i)9-s + (−0.215 + 0.976i)10-s + (0.779 + 0.626i)12-s + (−0.748 − 0.663i)13-s + (−0.644 − 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.836 + 0.548i)16-s + (−0.779 + 0.626i)17-s + (0.607 + 0.794i)18-s + ⋯
L(s)  = 1  + (0.989 + 0.144i)2-s + (0.926 + 0.377i)3-s + (0.958 + 0.285i)4-s + (−0.0724 + 0.997i)5-s + (0.861 + 0.506i)6-s + (−0.748 − 0.663i)7-s + (0.906 + 0.421i)8-s + (0.715 + 0.698i)9-s + (−0.215 + 0.976i)10-s + (0.779 + 0.626i)12-s + (−0.748 − 0.663i)13-s + (−0.644 − 0.764i)14-s + (−0.443 + 0.896i)15-s + (0.836 + 0.548i)16-s + (−0.779 + 0.626i)17-s + (0.607 + 0.794i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.997 - 0.0722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.997 - 0.0722i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (357, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.997 - 0.0722i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1062664766 + 2.939247789i\)
\(L(\frac12)\) \(\approx\) \(-0.1062664766 + 2.939247789i\)
\(L(1)\) \(\approx\) \(1.814243581 + 0.9685459108i\)
\(L(1)\) \(\approx\) \(1.814243581 + 0.9685459108i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.989 + 0.144i)T \)
3 \( 1 + (0.926 + 0.377i)T \)
5 \( 1 + (-0.0724 + 0.997i)T \)
7 \( 1 + (-0.748 - 0.663i)T \)
13 \( 1 + (-0.748 - 0.663i)T \)
17 \( 1 + (-0.779 + 0.626i)T \)
19 \( 1 + (0.998 + 0.0483i)T \)
23 \( 1 + (-0.485 + 0.873i)T \)
29 \( 1 + (-0.885 - 0.464i)T \)
31 \( 1 + (-0.981 + 0.192i)T \)
37 \( 1 + (-0.568 + 0.822i)T \)
41 \( 1 + (-0.262 - 0.964i)T \)
43 \( 1 + (-0.0724 + 0.997i)T \)
47 \( 1 + (-0.715 - 0.698i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.779 - 0.626i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.0724 + 0.997i)T \)
71 \( 1 + (0.681 - 0.732i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.644 + 0.764i)T \)
83 \( 1 + (-0.958 + 0.285i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (-0.399 + 0.916i)T \)
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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

−20.22084024616152207732237007711, −19.711220408499942588120247668, −18.92799710988095076268423461595, −18.1457780727613979367317378413, −16.817256431382532804636292997096, −16.05624847143412592691223339153, −15.61819422171725370011817131724, −14.64650565336277068121982723265, −14.009004178695853756885420129640, −13.20086129358973282383775367729, −12.628867736844708223550241031944, −12.11573830697613563426922115001, −11.280202390429034852171420005752, −9.80414876010926498911100486564, −9.36188702061473488132856382090, −8.521063245946587841139096733813, −7.4219471919195818871812363922, −6.82262245336417664228380330758, −5.78918153538180944645273875510, −4.92559216708805420814885682471, −4.08974185856500376290834132124, −3.213456737769713522675031423630, −2.31696054069661854366526654194, −1.645057112057678521836311723133, −0.2718381358797751163481655202, 1.73669777278092857683584261040, 2.621291830905414027710871195104, 3.512393359500427890981008741089, 3.73576618430796269668337339438, 4.932737879076236221501511573543, 5.91012640930023863008567123001, 6.97241197473594500347644518821, 7.38572618431802605240407247595, 8.17900422402150450023662991507, 9.59626160500928464478184502900, 10.15790308987904881004219986892, 10.893873111294869452006127600496, 11.770465197420149078828915360385, 12.9635478622643203729914422969, 13.36265408258742945336873003423, 14.18353325993405250930259272758, 14.754425622846487678841541720840, 15.49174311829317873676724094170, 15.9645461246923909358877017237, 16.955439385892131524833005348133, 17.8720600322858717357025883008, 19.057030651446610180297164255415, 19.69150769382246021076700479411, 20.121785949560197783665487829381, 20.94170709305610147342021878764

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