↑

Properties

Label 1-11e2-121.23-r0-0-0
Degree $1$
Conductor $121$
Sign $0.987 - 0.155i$
Analytic cond. $0.561921$
Root an. cond. $0.561921$
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  + (−0.654 − 0.755i)2-s + 3-s + (−0.142 + 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (−0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + 9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.415 + 0.909i)17-s + (−0.654 − 0.755i)18-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + 3-s + (−0.142 + 0.989i)4-s + (0.841 + 0.540i)5-s + (−0.654 − 0.755i)6-s + (−0.959 + 0.281i)7-s + (0.841 − 0.540i)8-s + 9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)14-s + (0.841 + 0.540i)15-s + (−0.959 − 0.281i)16-s + (0.415 + 0.909i)17-s + (−0.654 − 0.755i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.155i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.987 - 0.155i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.987 - 0.155i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.095315249 - 0.08548785493i\)
\(L(\frac12)\) \(\approx\) \(1.095315249 - 0.08548785493i\)
\(L(1)\) \(\approx\) \(1.065920126 - 0.1353916697i\)
\(L(1)\) \(\approx\) \(1.065920126 - 0.1353916697i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.654 - 0.755i)T \)
3 \( 1 + T \)
5 \( 1 + (0.841 + 0.540i)T \)
7 \( 1 + (-0.959 + 0.281i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.415 + 0.909i)T \)
19 \( 1 + (0.415 - 0.909i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
31 \( 1 + (-0.142 - 0.989i)T \)
37 \( 1 + (-0.142 - 0.989i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.654 + 0.755i)T \)
61 \( 1 + (-0.654 + 0.755i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (-0.959 - 0.281i)T \)
79 \( 1 + (0.841 + 0.540i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (0.841 - 0.540i)T \)
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

−29.138481738537808937982721959575, −27.782751449926275977435975569245, −26.875862858908321031669588154408, −25.770823561485826188376761092384, −25.27061137624111615617045405981, −24.547701395976082849085406619455, −23.24141140052067877183105967243, −21.98256699511352683115225421858, −20.43795859512311973016803104924, −19.94282254793600463453674668052, −18.68260115795202235000995186404, −17.79186890945645354499198367511, −16.44472354883433505076648276895, −15.81240996839055359221152807809, −14.40260540587622777164065367738, −13.661308820260919012758226121574, −12.58944990437184602676652824552, −10.13887904383238091858895091574, −9.779397935022180145829108024833, −8.62736171368410742632612723102, −7.5627155150018464322440328241, −6.31059657862548803120027231878, −5.00899208075965900996969326306, −3.111235111999360319315972843983, −1.39627214611297689115273971679, 1.911370602561225894551471679875, 2.82403449815562999378277504863, 4.03158691959458641970632474745, 6.35803752762531115242378490408, 7.5606148467480455339843342858, 9.025903398997397639095770862524, 9.62229136870484761238015056285, 10.57413621152143652889013658522, 12.18554483171024711568494436344, 13.28022194882900605472718295990, 14.086385640267014941390388800500, 15.54724026188949558946146430316, 16.777057542302487812486251585482, 18.03713007624857050283995680866, 19.01440435691604872197382757665, 19.55587242671601523474407284719, 20.84278840970923519901412476238, 21.66502282138584142660312746517, 22.40465495725739223947989347034, 24.30167652640428912901714545547, 25.60531757468948639462991973227, 26.01309796003230825585637902219, 26.707045560736755009529449929337, 28.20781073190838960819417439692, 29.02811949286429496336116261564

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