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Properties

Label 1-1441-1441.1088-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.678 - 0.734i$
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  + (0.715 − 0.698i)2-s + (0.644 + 0.764i)3-s + (0.0241 − 0.999i)4-s + (−0.527 + 0.849i)5-s + (0.995 + 0.0965i)6-s + (−0.399 − 0.916i)7-s + (−0.681 − 0.732i)8-s + (−0.168 + 0.985i)9-s + (0.215 + 0.976i)10-s + (0.779 − 0.626i)12-s + (−0.215 + 0.976i)13-s + (−0.926 − 0.377i)14-s + (−0.989 + 0.144i)15-s + (−0.998 − 0.0483i)16-s + (−0.262 − 0.964i)17-s + (0.568 + 0.822i)18-s + ⋯
L(s)  = 1  + (0.715 − 0.698i)2-s + (0.644 + 0.764i)3-s + (0.0241 − 0.999i)4-s + (−0.527 + 0.849i)5-s + (0.995 + 0.0965i)6-s + (−0.399 − 0.916i)7-s + (−0.681 − 0.732i)8-s + (−0.168 + 0.985i)9-s + (0.215 + 0.976i)10-s + (0.779 − 0.626i)12-s + (−0.215 + 0.976i)13-s + (−0.926 − 0.377i)14-s + (−0.989 + 0.144i)15-s + (−0.998 − 0.0483i)16-s + (−0.262 − 0.964i)17-s + (0.568 + 0.822i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.678 - 0.734i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.678 - 0.734i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1088, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.678 - 0.734i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5364643794 - 1.226838768i\)
\(L(\frac12)\) \(\approx\) \(0.5364643794 - 1.226838768i\)
\(L(1)\) \(\approx\) \(1.245701030 - 0.4172145181i\)
\(L(1)\) \(\approx\) \(1.245701030 - 0.4172145181i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.715 - 0.698i)T \)
3 \( 1 + (0.644 + 0.764i)T \)
5 \( 1 + (-0.527 + 0.849i)T \)
7 \( 1 + (-0.399 - 0.916i)T \)
13 \( 1 + (-0.215 + 0.976i)T \)
17 \( 1 + (-0.262 - 0.964i)T \)
19 \( 1 + (-0.354 - 0.935i)T \)
23 \( 1 + (-0.485 - 0.873i)T \)
29 \( 1 + (-0.168 - 0.985i)T \)
31 \( 1 + (0.906 - 0.421i)T \)
37 \( 1 + (-0.958 + 0.285i)T \)
41 \( 1 + (0.998 - 0.0483i)T \)
43 \( 1 + (0.0724 + 0.997i)T \)
47 \( 1 + (-0.885 - 0.464i)T \)
53 \( 1 + (0.309 - 0.951i)T \)
59 \( 1 + (0.836 - 0.548i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.0724 - 0.997i)T \)
71 \( 1 + (-0.120 - 0.992i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.970 - 0.239i)T \)
83 \( 1 + (-0.607 - 0.794i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.861 + 0.506i)T \)
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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

−21.0409268029456437472930625969, −20.26598034124946505755616119176, −19.54725773456845151729887686858, −18.86398387933138904088848602363, −17.76620185685514830750004507036, −17.32623570121439627801809121147, −16.181426570681856455018903529343, −15.5995166757495747841012046579, −14.97330801261256628168682033865, −14.25191870885958263859749851961, −13.20944607156438405549173436955, −12.686793865477123296135525999107, −12.29996736375206031465272570153, −11.533549127079472319747118522417, −10.01641209277408136709229708482, −8.77205495381493154944424899297, −8.53324405206857230345063730189, −7.7503344435080532935660313572, −6.95459308507365609280061593606, −5.842107527216543301562641976328, −5.491924535258528006293846836098, −4.14996151111919314044635539497, −3.454704923956971158637867108435, −2.56232654268288446389791511735, −1.45900168180605946070036947973, 0.337838551788001465067222323812, 2.123900166315096934488142540282, 2.774142489845682124810160512760, 3.61640221667004689322752735221, 4.336644817989568149282947329996, 4.81701610646783126668084535111, 6.362723952238699367266792148545, 6.923080134747849719882260962607, 7.945917899264696574396400386686, 9.133473190559521159367879155501, 9.86982294958599973176943748852, 10.43549280356282766559054545513, 11.28484095023562475779488891798, 11.72604932538973465714271828276, 13.061285458529724677509855780716, 13.733285327284324963091540791998, 14.29998380654371663835751301415, 14.91168382606844650147616336643, 15.823272473592960026265191157234, 16.24862931000669987264750515249, 17.4762383369062227321712414670, 18.61465998828449311097329460852, 19.370743022778537200278378849400, 19.644649533422765947778343560573, 20.55390419479697130000954694167

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