Properties

Label 1-1441-1441.344-r1-0-0
Degree $1$
Conductor $1441$
Sign $-0.999 + 0.00983i$
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  + (0.168 − 0.985i)2-s + (−0.0724 − 0.997i)3-s + (−0.943 − 0.331i)4-s + (0.644 + 0.764i)5-s + (−0.995 − 0.0965i)6-s + (−0.748 + 0.663i)7-s + (−0.485 + 0.873i)8-s + (−0.989 + 0.144i)9-s + (0.861 − 0.506i)10-s + (−0.262 + 0.964i)12-s + (−0.748 + 0.663i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (0.779 + 0.626i)16-s + (0.262 + 0.964i)17-s + (−0.0241 + 0.999i)18-s + ⋯
L(s)  = 1  + (0.168 − 0.985i)2-s + (−0.0724 − 0.997i)3-s + (−0.943 − 0.331i)4-s + (0.644 + 0.764i)5-s + (−0.995 − 0.0965i)6-s + (−0.748 + 0.663i)7-s + (−0.485 + 0.873i)8-s + (−0.989 + 0.144i)9-s + (0.861 − 0.506i)10-s + (−0.262 + 0.964i)12-s + (−0.748 + 0.663i)13-s + (0.527 + 0.849i)14-s + (0.715 − 0.698i)15-s + (0.779 + 0.626i)16-s + (0.262 + 0.964i)17-s + (−0.0241 + 0.999i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.999 + 0.00983i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ -0.999 + 0.00983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001485959712 - 0.3022170692i\)
\(L(\frac12)\) \(\approx\) \(0.001485959712 - 0.3022170692i\)
\(L(1)\) \(\approx\) \(0.6986999585 - 0.3500503874i\)
\(L(1)\) \(\approx\) \(0.6986999585 - 0.3500503874i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.168 + 0.985i)T \)
3 \( 1 + (0.0724 + 0.997i)T \)
5 \( 1 + (-0.644 - 0.764i)T \)
7 \( 1 + (0.748 - 0.663i)T \)
13 \( 1 + (0.748 - 0.663i)T \)
17 \( 1 + (-0.262 - 0.964i)T \)
19 \( 1 + (0.836 + 0.548i)T \)
23 \( 1 + (0.981 - 0.192i)T \)
29 \( 1 + (0.885 - 0.464i)T \)
31 \( 1 + (-0.681 - 0.732i)T \)
37 \( 1 + (0.568 + 0.822i)T \)
41 \( 1 + (0.998 - 0.0483i)T \)
43 \( 1 + (-0.644 - 0.764i)T \)
47 \( 1 + (-0.989 + 0.144i)T \)
53 \( 1 + (-0.309 - 0.951i)T \)
59 \( 1 + (0.262 + 0.964i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.644 - 0.764i)T \)
71 \( 1 + (-0.906 + 0.421i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.527 + 0.849i)T \)
83 \( 1 + (-0.943 + 0.331i)T \)
89 \( 1 + (0.809 + 0.587i)T \)
97 \( 1 + (0.215 - 0.976i)T \)
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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

−20.876036697494898968719384232104, −20.45707906997088871233415919134, −19.50448617476110931938668105337, −18.438107518693244750480883128665, −17.42730606939507739972172371313, −16.84432828519575487070020934431, −16.59592773249956661247366611084, −15.632766927696535031386980611604, −15.10866594714429994992522565601, −13.97117483744383312991826716954, −13.6944124091766080097201907671, −12.63137474486415756863657629390, −12.00276580181226285777222583404, −10.48178450955131932489327440554, −9.85372546531880067729449144115, −9.44174491537643411334503877699, −8.466496642888951965162933544157, −7.71636766223077047611166165049, −6.579567692469433900186976532142, −5.80421641445518467543655198538, −5.14894435270314935708124915111, −4.32926211421255961110995776307, −3.63530348019014266491785174479, −2.48586441822888701655366885116, −0.62839904211913993616376320917, 0.080895612363896841248090333855, 1.60384895903085942225029590724, 2.19569453002669224684152634403, 2.87844948617665538637948141579, 3.81760974084466513251919873201, 5.176386101644678591401941925596, 6.01886060718208330121194484536, 6.55081727921044114611203102276, 7.60518360873574215499081716945, 8.75501699018908072806781320753, 9.33758919096492936014047899072, 10.299586613472935115061060192469, 10.93615960668453209230691007814, 11.97341618611714174329668807171, 12.41559363272483794444165893855, 13.1715501635758769163068914588, 13.884052286293501008243614867992, 14.5465782765235396994307196924, 15.285252852461074997216714837321, 16.77814017636763015513433228087, 17.44385347627001677874045426356, 18.15441129224480026541993891059, 18.82724923063624422159075970382, 19.35396168807791469714698617640, 19.809897689629352315874479805369

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