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Properties

Label 1-1441-1441.227-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.760 + 0.649i$
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  + (−0.262 + 0.964i)2-s + (0.0241 + 0.999i)3-s + (−0.861 − 0.506i)4-s + (0.958 + 0.285i)5-s + (−0.970 − 0.239i)6-s + (0.0724 − 0.997i)7-s + (0.715 − 0.698i)8-s + (−0.998 + 0.0483i)9-s + (−0.527 + 0.849i)10-s + (0.485 − 0.873i)12-s + (−0.644 + 0.764i)13-s + (0.943 + 0.331i)14-s + (−0.262 + 0.964i)15-s + (0.485 + 0.873i)16-s + (0.120 − 0.992i)17-s + (0.215 − 0.976i)18-s + ⋯
L(s)  = 1  + (−0.262 + 0.964i)2-s + (0.0241 + 0.999i)3-s + (−0.861 − 0.506i)4-s + (0.958 + 0.285i)5-s + (−0.970 − 0.239i)6-s + (0.0724 − 0.997i)7-s + (0.715 − 0.698i)8-s + (−0.998 + 0.0483i)9-s + (−0.527 + 0.849i)10-s + (0.485 − 0.873i)12-s + (−0.644 + 0.764i)13-s + (0.943 + 0.331i)14-s + (−0.262 + 0.964i)15-s + (0.485 + 0.873i)16-s + (0.120 − 0.992i)17-s + (0.215 − 0.976i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.760 + 0.649i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.760 + 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.156980253 + 0.4268133745i\)
\(L(\frac12)\) \(\approx\) \(1.156980253 + 0.4268133745i\)
\(L(1)\) \(\approx\) \(0.8268790252 + 0.4837667903i\)
\(L(1)\) \(\approx\) \(0.8268790252 + 0.4837667903i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (-0.262 + 0.964i)T \)
3 \( 1 + (0.0241 + 0.999i)T \)
5 \( 1 + (0.958 + 0.285i)T \)
7 \( 1 + (0.0724 - 0.997i)T \)
13 \( 1 + (-0.644 + 0.764i)T \)
17 \( 1 + (0.120 - 0.992i)T \)
19 \( 1 + (0.485 - 0.873i)T \)
23 \( 1 + (-0.715 - 0.698i)T \)
29 \( 1 + (0.779 + 0.626i)T \)
31 \( 1 + (-0.715 + 0.698i)T \)
37 \( 1 + (-0.399 - 0.916i)T \)
41 \( 1 + (-0.120 - 0.992i)T \)
43 \( 1 + (0.943 - 0.331i)T \)
47 \( 1 + (0.262 - 0.964i)T \)
53 \( 1 + (-0.809 - 0.587i)T \)
59 \( 1 + (0.981 - 0.192i)T \)
61 \( 1 - T \)
67 \( 1 + (0.943 + 0.331i)T \)
71 \( 1 + (-0.715 + 0.698i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.958 + 0.285i)T \)
83 \( 1 + (0.399 - 0.916i)T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.0724 - 0.997i)T \)
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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

−20.60573141235757034759553855318, −19.78993425916566159829591756907, −19.12520952636823902459450419761, −18.4049783898912307587942617858, −17.78026541526835515772041652288, −17.339152431362646687556468059809, −16.4846457733736841568615039667, −15.09316281459528610283561108441, −14.25837333573736375267211598490, −13.57881604122094125559769092094, −12.72982141661199346815824018334, −12.36961531775868394881462423806, −11.66683826106656760248411475186, −10.626945349768279014290919337137, −9.756940961000917947903788089814, −9.15798531538690081186073026109, −8.11704623465768646979619257447, −7.80835442658865046375469120855, −6.204817425190988842333250670379, −5.70901793032859065719606367225, −4.832629545637630825464862020995, −3.378451222186006527725241820018, −2.52898596713668655493726889946, −1.87561625067673764517411688078, −1.10375110804781144150793405536, 0.56515310527028283920476080633, 2.07755604674907160998272292088, 3.30777443903334646338499127996, 4.361886548656429754375178433176, 5.00283964746174437683301153414, 5.71373662304804908749888805244, 6.9013168274937804722390690211, 7.1892908517976042519568480443, 8.56687864802818753118744229405, 9.23512231344044413391833962551, 9.90025869341838147190573245713, 10.455850248805029370592049023101, 11.25541652539912819443767262149, 12.59114301301655282962595596386, 13.88168193762270678106096515869, 14.03622150198895273929870172842, 14.59880443864994854353747372297, 15.76043031568440895921886633361, 16.28628617567007065139765851480, 16.95922859166041824581697124420, 17.60864803971772111589467108562, 18.17423265564130629024306455803, 19.31604644022781497427712246979, 20.08930780999859407729021507814, 20.855824508989310137298171599637

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