Properties

Label 1-1441-1441.1026-r0-0-0
Degree $1$
Conductor $1441$
Sign $-0.606 + 0.794i$
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  + (−0.0724 − 0.997i)2-s + (0.981 − 0.192i)3-s + (−0.989 + 0.144i)4-s + (−0.681 + 0.732i)5-s + (−0.262 − 0.964i)6-s + (−0.354 − 0.935i)7-s + (0.215 + 0.976i)8-s + (0.926 − 0.377i)9-s + (0.779 + 0.626i)10-s + (−0.943 + 0.331i)12-s + (−0.354 − 0.935i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (0.958 − 0.285i)16-s + (−0.943 − 0.331i)17-s + (−0.443 − 0.896i)18-s + ⋯
L(s)  = 1  + (−0.0724 − 0.997i)2-s + (0.981 − 0.192i)3-s + (−0.989 + 0.144i)4-s + (−0.681 + 0.732i)5-s + (−0.262 − 0.964i)6-s + (−0.354 − 0.935i)7-s + (0.215 + 0.976i)8-s + (0.926 − 0.377i)9-s + (0.779 + 0.626i)10-s + (−0.943 + 0.331i)12-s + (−0.354 − 0.935i)13-s + (−0.906 + 0.421i)14-s + (−0.527 + 0.849i)15-s + (0.958 − 0.285i)16-s + (−0.943 − 0.331i)17-s + (−0.443 − 0.896i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.606 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $-0.606 + 0.794i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (1026, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ -0.606 + 0.794i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1872375331 - 0.3784106982i\)
\(L(\frac12)\) \(\approx\) \(-0.1872375331 - 0.3784106982i\)
\(L(1)\) \(\approx\) \(0.7050058696 - 0.4992136076i\)
\(L(1)\) \(\approx\) \(0.7050058696 - 0.4992136076i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.0724 - 0.997i)T \)
3 \( 1 + (0.981 - 0.192i)T \)
5 \( 1 + (-0.681 + 0.732i)T \)
7 \( 1 + (-0.354 - 0.935i)T \)
13 \( 1 + (-0.354 - 0.935i)T \)
17 \( 1 + (-0.943 - 0.331i)T \)
19 \( 1 + (0.0241 + 0.999i)T \)
23 \( 1 + (-0.861 - 0.506i)T \)
29 \( 1 + (-0.970 + 0.239i)T \)
31 \( 1 + (0.995 + 0.0965i)T \)
37 \( 1 + (0.885 + 0.464i)T \)
41 \( 1 + (-0.607 - 0.794i)T \)
43 \( 1 + (-0.681 + 0.732i)T \)
47 \( 1 + (0.926 - 0.377i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (-0.943 - 0.331i)T \)
61 \( 1 + (-0.809 + 0.587i)T \)
67 \( 1 + (-0.681 - 0.732i)T \)
71 \( 1 + (0.399 - 0.916i)T \)
73 \( 1 + (-0.809 + 0.587i)T \)
79 \( 1 + (-0.906 - 0.421i)T \)
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

−21.513069688881487166488578375497, −20.21715060050800360676374212958, −19.60558663029817362107339796121, −19.00181533383566350181426592968, −18.308602867060724856970912166178, −17.27009312659532223488695777454, −16.43641141058830491413325038502, −15.716250861878872468121223841746, −15.34305866003247649151298734075, −14.67435490664682143507268865037, −13.60202075109887549244734567704, −13.148904908947720000035955863844, −12.26961399896845649315572377766, −11.363830909540831463500343880370, −9.90393829758042721550202094632, −9.23679406344646235625588773964, −8.781738042256797254850505705375, −8.08702847414657297337753219551, −7.26619924020886813291658318345, −6.4362437147507538090823394432, −5.336413642659438642963988010448, −4.44384807661485391484044651272, −3.95051327525529726842908956985, −2.72244090386139913131891308906, −1.603273986885339429930607708842, 0.14265332063145642344451133366, 1.44330264097469293373048092464, 2.58399135670621894805869731993, 3.16608383535937420497137661305, 3.97799769166636618128089100071, 4.53593071962468038844362073554, 6.13682176945160630141458453144, 7.25330378350575932951764974973, 7.83571927513310336038649685165, 8.52961269628225039015570780705, 9.62373031672959908314537618280, 10.27672290391045131721752968790, 10.78410359224060766362512817070, 11.89541186041879078219159982789, 12.56655200481234243882063919817, 13.443621859688225063775576882574, 13.95257331695627389878479179317, 14.76836175361751021067706111592, 15.446452727622939866883994861613, 16.50707247581700439737058455736, 17.524795597447994089505137703642, 18.4342108125395988784819978255, 18.7709669456030479328372960039, 19.870526537719074849449256203677, 20.00066547137325868603925923820

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