↑

Properties

Label 1-11e2-121.86-r0-0-0
Degree $1$
Conductor $121$
Sign $0.157 - 0.987i$
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  + (0.0855 + 0.996i)2-s + (0.309 − 0.951i)3-s + (−0.985 + 0.170i)4-s + (−0.998 + 0.0570i)5-s + (0.974 + 0.226i)6-s + (−0.564 − 0.825i)7-s + (−0.254 − 0.967i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (0.696 − 0.717i)13-s + (0.774 − 0.633i)14-s + (−0.254 + 0.967i)15-s + (0.941 − 0.336i)16-s + (−0.870 − 0.491i)17-s + (0.516 − 0.856i)18-s + ⋯
L(s)  = 1  + (0.0855 + 0.996i)2-s + (0.309 − 0.951i)3-s + (−0.985 + 0.170i)4-s + (−0.998 + 0.0570i)5-s + (0.974 + 0.226i)6-s + (−0.564 − 0.825i)7-s + (−0.254 − 0.967i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)10-s + (−0.142 + 0.989i)12-s + (0.696 − 0.717i)13-s + (0.774 − 0.633i)14-s + (−0.254 + 0.967i)15-s + (0.941 − 0.336i)16-s + (−0.870 − 0.491i)17-s + (0.516 − 0.856i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.157 - 0.987i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (86, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.157 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4604859668 - 0.3927755287i\)
\(L(\frac12)\) \(\approx\) \(0.4604859668 - 0.3927755287i\)
\(L(1)\) \(\approx\) \(0.7463533312 - 0.07946492925i\)
\(L(1)\) \(\approx\) \(0.7463533312 - 0.07946492925i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.0855 + 0.996i)T \)
3 \( 1 + (0.309 - 0.951i)T \)
5 \( 1 + (-0.998 + 0.0570i)T \)
7 \( 1 + (-0.564 - 0.825i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (-0.870 - 0.491i)T \)
19 \( 1 + (-0.736 - 0.676i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (0.993 + 0.113i)T \)
31 \( 1 + (-0.466 + 0.884i)T \)
37 \( 1 + (0.897 - 0.441i)T \)
41 \( 1 + (-0.921 + 0.389i)T \)
43 \( 1 + (0.841 - 0.540i)T \)
47 \( 1 + (0.516 + 0.856i)T \)
53 \( 1 + (0.941 + 0.336i)T \)
59 \( 1 + (-0.921 - 0.389i)T \)
61 \( 1 + (0.0855 - 0.996i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (-0.0285 - 0.999i)T \)
79 \( 1 + (-0.362 - 0.931i)T \)
83 \( 1 + (0.610 - 0.791i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
97 \( 1 + (-0.998 - 0.0570i)T \)
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

−28.97884455452108753061442077454, −28.18752345731029154437191347679, −27.51906497034454986397767942470, −26.52349189431747482453674432058, −25.63987942840640626950755962452, −23.89576430939560117867654692617, −22.84874360675127775781403120490, −21.981127279760883535192814184347, −21.16078978645647793669151539520, −20.04240362525973163328587590533, −19.34049583435334560839377803385, −18.38376748470977104076320466937, −16.70070382429494946943429101266, −15.62723793221038415078228523456, −14.75831578037066359048526185109, −13.42007260310521207552360640441, −12.12524310809316439711275043392, −11.277689076728095769552288544057, −10.18433131570465480756744052638, −8.96719097476941997139349913574, −8.30346448671685493446051232102, −5.96975188110141348877905133250, −4.384962622395362947671905063302, −3.68649766873340199473851467605, −2.33221206353971570349187916518, 0.53439310678047814006243846623, 3.18647804344691980305846276820, 4.39843141002281305092569539640, 6.250650571285617693176318223934, 7.08615333668969273168973703079, 8.014281777599610269410915725971, 8.95986786264213486464117191425, 10.752582841439355645317983584, 12.33734967610035448952835390469, 13.22890392378429689521374754589, 14.1244706419627290074428254605, 15.37123166065305359225160486204, 16.22615291345243889317113794996, 17.50555922192282389576841413490, 18.39139817577740335596727936512, 19.525695309987681215860681479930, 20.23900127760781334480176402493, 22.2232012371590214110518781959, 23.30725171584913846653019103224, 23.602085970789553188081871595114, 24.74881067195989502616181060563, 25.747300669862537258733259359914, 26.525589831874273757995403561608, 27.528867656966850282235143952558, 28.809158233094340366754462421513

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