Properties

Label 1-11e2-121.38-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

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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: $\chi_{121} (38, \cdot )$
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 \( 1 + (-0.998 + 0.0570i)T \)
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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

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

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