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Properties

Label 1-1441-1441.10-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.666 - 0.745i$
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  + (0.644 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.666 - 0.745i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.666 - 0.745i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.838956275 - 1.270569636i\)
\(L(\frac12)\) \(\approx\) \(2.838956275 - 1.270569636i\)
\(L(1)\) \(\approx\) \(1.829612592 - 0.7045214315i\)
\(L(1)\) \(\approx\) \(1.829612592 - 0.7045214315i\)

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

−20.66095999065042987537422144792, −20.31584127908627017964856689164, −19.39287184773854474807187667545, −18.36386513474418524168205168340, −18.0479824799123475865935762791, −16.814913473002822998831998442605, −16.007371962853056809407227306187, −15.49552884343890697247674256743, −14.53802382648817492881640537812, −14.15955710683262169590288961917, −13.5393303736013608655324095883, −12.68518332466455681489127244844, −11.79778161550070255370448223272, −10.89612248116560566947729278636, −9.99648404810570501040501655455, −8.79516519561639479741268596080, −8.15669645902277809897974513514, −7.37047147716053307457854715135, −7.001014475057410735530287968286, −6.035101407788749686498759169150, −4.69269874056170179395674927049, −3.966439227628075519714590973137, −3.323501709002563096012187634142, −2.58778006787821381472526676071, −1.00271244522003351716751771678, 1.23004812886009458669643044108, 1.85991491950381030826728409419, 3.100260764425030570110734779909, 3.63948844762106518958030396111, 4.46041487986696501703590875980, 5.37977383538786677316710836611, 6.10307359104334470471477413419, 7.685533949287850472740494195010, 8.23016474923868013998431777024, 9.192003706554914995803009406147, 9.600137732661983083878355784478, 10.76615257026665335740118065676, 11.55733742329254788790636294349, 12.422205291693663540121967917015, 12.7774050003057329767658588011, 13.882069220830825283468582708014, 14.35894105208878984945453579423, 15.340565034602738530534711595797, 15.726676753997011972818850697529, 16.508572513185910913469477333648, 18.10559318154844091229961227518, 18.607776945472435791041882014855, 19.39553928050659488729835330346, 19.9398655371478572893299044072, 20.72021607783530832229607841310

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