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Properties

Label 1-1441-1441.406-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.988 - 0.152i$
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  + (−0.644 − 0.764i)2-s + (0.981 − 0.192i)3-s + (−0.168 + 0.985i)4-s + (−0.681 + 0.732i)5-s + (−0.779 − 0.626i)6-s + (0.262 − 0.964i)7-s + (0.861 − 0.506i)8-s + (0.926 − 0.377i)9-s + (0.998 + 0.0483i)10-s + (0.0241 + 0.999i)12-s + (0.998 − 0.0483i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (−0.943 − 0.331i)16-s + (−0.958 + 0.285i)17-s + (−0.885 − 0.464i)18-s + ⋯
L(s)  = 1  + (−0.644 − 0.764i)2-s + (0.981 − 0.192i)3-s + (−0.168 + 0.985i)4-s + (−0.681 + 0.732i)5-s + (−0.779 − 0.626i)6-s + (0.262 − 0.964i)7-s + (0.861 − 0.506i)8-s + (0.926 − 0.377i)9-s + (0.998 + 0.0483i)10-s + (0.0241 + 0.999i)12-s + (0.998 − 0.0483i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (−0.943 − 0.331i)16-s + (−0.958 + 0.285i)17-s + (−0.885 − 0.464i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.988 - 0.152i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.988 - 0.152i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (406, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.988 - 0.152i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07475723105 - 0.9730372523i\)
\(L(\frac12)\) \(\approx\) \(0.07475723105 - 0.9730372523i\)
\(L(1)\) \(\approx\) \(0.8383744902 - 0.3591786982i\)
\(L(1)\) \(\approx\) \(0.8383744902 - 0.3591786982i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.644 - 0.764i)T \)
3 \( 1 + (0.981 - 0.192i)T \)
5 \( 1 + (-0.681 + 0.732i)T \)
7 \( 1 + (0.262 - 0.964i)T \)
13 \( 1 + (0.998 - 0.0483i)T \)
17 \( 1 + (-0.958 + 0.285i)T \)
19 \( 1 + (-0.568 + 0.822i)T \)
23 \( 1 + (0.399 + 0.916i)T \)
29 \( 1 + (-0.926 - 0.377i)T \)
31 \( 1 + (0.995 + 0.0965i)T \)
37 \( 1 + (-0.443 - 0.896i)T \)
41 \( 1 + (0.943 - 0.331i)T \)
43 \( 1 + (-0.485 - 0.873i)T \)
47 \( 1 + (-0.970 - 0.239i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (-0.607 + 0.794i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (0.485 - 0.873i)T \)
71 \( 1 + (-0.748 - 0.663i)T \)
73 \( 1 + (-0.309 - 0.951i)T \)
79 \( 1 + (-0.120 + 0.992i)T \)
83 \( 1 + (0.989 + 0.144i)T \)
89 \( 1 + (-0.809 + 0.587i)T \)
97 \( 1 + (0.836 + 0.548i)T \)
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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

−20.58050453452209234553888083494, −20.137795431097528566748695745147, −19.183309482013461356048438289994, −18.78885023672508241720554451862, −17.97868519594577281261866726695, −17.0514496501611906997045955471, −16.00996463257960970629207027310, −15.72366855441551759737308903439, −15.067048773344178735141077094905, −14.38119591457798779787055977581, −13.299520842426218422368247484370, −12.82410789748316224606049688583, −11.45541614811697960145699640902, −10.880211178194102053446326238486, −9.61460318936395081091697021765, −8.966591560195454697927295706396, −8.49695430948982470132169961559, −7.99902555489961209955349824607, −6.927156020906335395127733082302, −6.10581486664763701755867274380, −4.768945184829143211475730388607, −4.53830525468842185326042731679, −3.12323436410711375192519460864, −2.04204950736527633302301568437, −1.10658149908409776092196888989, 0.21112030101366673588516396756, 1.38847899021439964637876752874, 2.16208226704294581912420258152, 3.32275235786330435372145222358, 3.785386725145032176098834784479, 4.43550618488442958508563521435, 6.38497027005014708894241270257, 7.20647234885511600079318821193, 7.83555778215451649833506661341, 8.44554596766312953505765457776, 9.28423389145118786150748282968, 10.2943396467709957915023091775, 10.810451265612050314104481579191, 11.51541659953763719558957125029, 12.53054950187494489201654313432, 13.40784609693718217520635045869, 13.84502081182521586109643970387, 14.870920923109248583421105371414, 15.616651629291222979446173876489, 16.43280267240554662950626165555, 17.49677385182661847363145933247, 18.11462253835746664699069318017, 18.90410377808496654683499069168, 19.49998246264152846064060311557, 19.9423191533145294718662202876

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