↑

Properties

Label 1-1441-1441.570-r0-0-0
Degree $1$
Conductor $1441$
Sign $0.488 - 0.872i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.215 − 0.976i)2-s + (0.836 + 0.548i)3-s + (−0.906 − 0.421i)4-s + (0.779 − 0.626i)5-s + (0.715 − 0.698i)6-s + (0.885 − 0.464i)7-s + (−0.607 + 0.794i)8-s + (0.399 + 0.916i)9-s + (−0.443 − 0.896i)10-s + (−0.527 − 0.849i)12-s + (0.885 − 0.464i)13-s + (−0.262 − 0.964i)14-s + (0.995 − 0.0965i)15-s + (0.644 + 0.764i)16-s + (−0.527 + 0.849i)17-s + (0.981 − 0.192i)18-s + ⋯
L(s)  = 1  + (0.215 − 0.976i)2-s + (0.836 + 0.548i)3-s + (−0.906 − 0.421i)4-s + (0.779 − 0.626i)5-s + (0.715 − 0.698i)6-s + (0.885 − 0.464i)7-s + (−0.607 + 0.794i)8-s + (0.399 + 0.916i)9-s + (−0.443 − 0.896i)10-s + (−0.527 − 0.849i)12-s + (0.885 − 0.464i)13-s + (−0.262 − 0.964i)14-s + (0.995 − 0.0965i)15-s + (0.644 + 0.764i)16-s + (−0.527 + 0.849i)17-s + (0.981 − 0.192i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.488 - 0.872i$
Analytic conductor: \(6.69197\)
Root analytic conductor: \(6.69197\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (570, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (0:\ ),\ 0.488 - 0.872i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.524057391 - 1.480552488i\)
\(L(\frac12)\) \(\approx\) \(2.524057391 - 1.480552488i\)
\(L(1)\) \(\approx\) \(1.666895207 - 0.7287003012i\)
\(L(1)\) \(\approx\) \(1.666895207 - 0.7287003012i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.215 - 0.976i)T \)
3 \( 1 + (0.836 + 0.548i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (0.885 - 0.464i)T \)
13 \( 1 + (0.885 - 0.464i)T \)
17 \( 1 + (-0.527 + 0.849i)T \)
19 \( 1 + (-0.0724 + 0.997i)T \)
23 \( 1 + (0.0241 + 0.999i)T \)
29 \( 1 + (-0.748 - 0.663i)T \)
31 \( 1 + (0.958 - 0.285i)T \)
37 \( 1 + (0.120 - 0.992i)T \)
41 \( 1 + (0.926 + 0.377i)T \)
43 \( 1 + (0.779 - 0.626i)T \)
47 \( 1 + (0.399 + 0.916i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.527 + 0.849i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.779 + 0.626i)T \)
71 \( 1 + (-0.943 - 0.331i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (-0.262 + 0.964i)T \)
83 \( 1 + (-0.906 + 0.421i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.168 - 0.985i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.999683284883400536005514463987, −20.24917909608622504454405887554, −18.79881064668844979273517073654, −18.6220091208603243654925654922, −17.7837758150792777214558915712, −17.39295069548562511482627720326, −16.10435612173708360310821812068, −15.39012837097678489629628115928, −14.62567391293667688947229953838, −14.14880200175686640802311982259, −13.50749722315361641500607871261, −12.887008166247271441780926462603, −11.7913754678310131709642669104, −10.88268764866021101524402249512, −9.626992664848604599325612051684, −8.938694870167455150758095891514, −8.441442764522585241886320313296, −7.43454902186068710249229968763, −6.73468245305314089181605944360, −6.14902056382574949308871897578, −5.07712539160349707158104473629, −4.22667942223183721574556584292, −3.022174937828206402881450316772, −2.32356788415286219771798548885, −1.153026250913281622166366115933, 1.22251666197621441054921009746, 1.79866170728244901222042259787, 2.72032755351211029658988830048, 4.003154359871812347602167911142, 4.20385740135818232864431409053, 5.37881678882629743108107333014, 5.97513809401643278050622869185, 7.81250374617810715106816417000, 8.30626738898001772165276595720, 9.16528896558676320845812579332, 9.77250343643329203252040267935, 10.68392122271564001556143015985, 11.05382980932702873768590629448, 12.32786343753694834115426639911, 13.13421948926289481368055710024, 13.69339181822672852049509943946, 14.2650360033282038100677385903, 15.06371163085592749233678823446, 15.93904485197197776548927977779, 17.12354707135494502204641687355, 17.577054953120909233866838289403, 18.50511163711141141334067025680, 19.358776804028688276542349097905, 20.10953767341715814981540019261, 20.826808709131388442141015975633

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