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Properties

Label 1-1441-1441.29-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.344 - 0.938i$
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  + (0.836 + 0.548i)2-s + (0.568 − 0.822i)3-s + (0.399 + 0.916i)4-s + (0.568 − 0.822i)5-s + (0.926 − 0.377i)6-s + (−0.926 − 0.377i)7-s + (−0.168 + 0.985i)8-s + (−0.354 − 0.935i)9-s + (0.926 − 0.377i)10-s + (0.981 + 0.192i)12-s + (0.0724 − 0.997i)13-s + (−0.568 − 0.822i)14-s + (−0.354 − 0.935i)15-s + (−0.681 + 0.732i)16-s + (0.485 + 0.873i)17-s + (0.215 − 0.976i)18-s + ⋯
L(s)  = 1  + (0.836 + 0.548i)2-s + (0.568 − 0.822i)3-s + (0.399 + 0.916i)4-s + (0.568 − 0.822i)5-s + (0.926 − 0.377i)6-s + (−0.926 − 0.377i)7-s + (−0.168 + 0.985i)8-s + (−0.354 − 0.935i)9-s + (0.926 − 0.377i)10-s + (0.981 + 0.192i)12-s + (0.0724 − 0.997i)13-s + (−0.568 − 0.822i)14-s + (−0.354 − 0.935i)15-s + (−0.681 + 0.732i)16-s + (0.485 + 0.873i)17-s + (0.215 − 0.976i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.344 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.344 - 0.938i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.344 - 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.436133512 - 1.701253483i\)
\(L(\frac12)\) \(\approx\) \(2.436133512 - 1.701253483i\)
\(L(1)\) \(\approx\) \(1.916337071 - 0.4048986202i\)
\(L(1)\) \(\approx\) \(1.916337071 - 0.4048986202i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.836 + 0.548i)T \)
3 \( 1 + (0.568 - 0.822i)T \)
5 \( 1 + (0.568 - 0.822i)T \)
7 \( 1 + (-0.926 - 0.377i)T \)
13 \( 1 + (0.0724 - 0.997i)T \)
17 \( 1 + (0.485 + 0.873i)T \)
19 \( 1 + (0.485 - 0.873i)T \)
23 \( 1 + (0.989 + 0.144i)T \)
29 \( 1 + (0.836 - 0.548i)T \)
31 \( 1 + (-0.885 - 0.464i)T \)
37 \( 1 + (-0.995 + 0.0965i)T \)
41 \( 1 + (-0.981 - 0.192i)T \)
43 \( 1 + (0.607 + 0.794i)T \)
47 \( 1 + (0.262 - 0.964i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.906 - 0.421i)T \)
61 \( 1 + (0.809 - 0.587i)T \)
67 \( 1 + (0.607 - 0.794i)T \)
71 \( 1 + (-0.715 + 0.698i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.958 + 0.285i)T \)
83 \( 1 + (-0.748 - 0.663i)T \)
89 \( 1 + (0.309 - 0.951i)T \)
97 \( 1 + (0.970 - 0.239i)T \)
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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

−20.947808042373984376876656119905, −20.42768593583542931280851410551, −19.27555259091105730063310539613, −19.01223785663351594368332660373, −18.21868914269071459662502694932, −16.7794367348059128252683719099, −16.08129046456663526921619609740, −15.4750581469186794224235442048, −14.50535584430841644850295197328, −14.14341336778031005895968211013, −13.49628428142496397935348672360, −12.5103281781960649762735273247, −11.659077555849968984411262392281, −10.77712815737653064023735126683, −10.13259676358045089967887204331, −9.48367058695926919651729749867, −8.8957784653755880865318622566, −7.272092453655571517547423299479, −6.62224582600502037324393761408, −5.6188727127830845196092423886, −4.99264742510948023945547525849, −3.78484049506220512422840440825, −3.19398358705429386369114546655, −2.58068859964963563142054453883, −1.59184833857779136282862892029, 0.74339385165539448060906409583, 1.94906260419533915087355686527, 3.026474470790208411664506201333, 3.56430472598369286315685483624, 4.79782031036740492150787484651, 5.69913219310614111143655663640, 6.3474496496404195295165014013, 7.1673249251314530494127582271, 7.9555321647405903472135980875, 8.731507973993965339276588385450, 9.49799792320741973257468731656, 10.58365335144505723699475166846, 11.83565763438679287981013495973, 12.68760713950734597280322844038, 12.977705933627461546817832487115, 13.59433208614917930429108094226, 14.282777865413973813036733727441, 15.31538255779498500747310963609, 15.82042367076646267302383317878, 17.06733062321862846374332054888, 17.18741714798425263891462345863, 18.18610662916047708121839749425, 19.284316781736860137009582004968, 20.02505958926653972051449644206, 20.50805485392032565953791426164

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