L(s) = 1 | + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)11-s + (0.142 + 0.989i)13-s + (−0.415 − 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.841 − 0.540i)19-s + (−0.959 − 0.281i)20-s + (−0.959 + 0.281i)22-s + (0.959 + 0.281i)23-s + ⋯ |
L(s) = 1 | + (0.841 + 0.540i)2-s + (0.415 + 0.909i)4-s + (−0.654 + 0.755i)5-s + (−0.841 − 0.540i)7-s + (−0.142 + 0.989i)8-s + (−0.959 + 0.281i)10-s + (−0.654 + 0.755i)11-s + (0.142 + 0.989i)13-s + (−0.415 − 0.909i)14-s + (−0.654 + 0.755i)16-s + (−0.415 + 0.909i)17-s + (0.841 − 0.540i)19-s + (−0.959 − 0.281i)20-s + (−0.959 + 0.281i)22-s + (0.959 + 0.281i)23-s + ⋯ |
Λ(s)=(=(201s/2ΓR(s)L(s)(−0.687+0.725i)Λ(1−s)
Λ(s)=(=(201s/2ΓR(s)L(s)(−0.687+0.725i)Λ(1−s)
Degree: |
1 |
Conductor: |
201
= 3⋅67
|
Sign: |
−0.687+0.725i
|
Analytic conductor: |
0.933440 |
Root analytic conductor: |
0.933440 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ201(137,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 201, (0: ), −0.687+0.725i)
|
Particular Values
L(21) |
≈ |
0.5258615849+1.222942315i |
L(21) |
≈ |
0.5258615849+1.222942315i |
L(1) |
≈ |
1.026372748+0.7656479065i |
L(1) |
≈ |
1.026372748+0.7656479065i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 67 | 1 |
good | 2 | 1+(0.841+0.540i)T |
| 5 | 1+(−0.654+0.755i)T |
| 7 | 1+(−0.841−0.540i)T |
| 11 | 1+(−0.654+0.755i)T |
| 13 | 1+(0.142+0.989i)T |
| 17 | 1+(−0.415+0.909i)T |
| 19 | 1+(0.841−0.540i)T |
| 23 | 1+(0.959+0.281i)T |
| 29 | 1−T |
| 31 | 1+(0.142−0.989i)T |
| 37 | 1+T |
| 41 | 1+(0.415−0.909i)T |
| 43 | 1+(−0.415+0.909i)T |
| 47 | 1+(0.959+0.281i)T |
| 53 | 1+(0.415+0.909i)T |
| 59 | 1+(0.142−0.989i)T |
| 61 | 1+(0.654+0.755i)T |
| 71 | 1+(−0.415−0.909i)T |
| 73 | 1+(−0.654−0.755i)T |
| 79 | 1+(0.142+0.989i)T |
| 83 | 1+(0.654−0.755i)T |
| 89 | 1+(0.959−0.281i)T |
| 97 | 1−T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−26.7552553719916532284543286328, −25.08422716582236860336547551125, −24.66293352015286612652885278493, −23.45727070706359065874633886476, −22.78578150380048621109015587447, −21.87814062379648444458741050272, −20.72289635166183369960480044000, −20.11611071390770150261096896034, −19.097602589522889969584821922004, −18.29834448455303035724597625733, −16.401566510634703654893522127907, −15.85264206681340921931761620714, −14.963981686713909485150165698339, −13.45782949000260443809598550639, −12.89283413412035468400029451306, −11.94849711661368708873422016674, −10.9990055117642095610066344508, −9.76250639861285815040398301617, −8.65745990924392822759376007387, −7.247576129392121597160644562508, −5.755607166650076407420323975500, −5.05984178361529798230220819780, −3.575011137983380617559283541870, −2.75083019888413403885742086030, −0.776089820147681477182095972814,
2.47281155441932568263282125741, 3.636565022930741865750870959589, 4.508756878980769056623078816033, 6.069638926258411362776465027, 7.06771046111135330817956475979, 7.65685259591740330785558883197, 9.27395406406590113187570111861, 10.71687739704950325120953677333, 11.61823419549539959101517381332, 12.82399660785939857120476146181, 13.580464500606299281959339193217, 14.7790882872251403655662732206, 15.49466074673483640341355608395, 16.37824184853399008692804918075, 17.410919377479772291128873740127, 18.67989398753253127725972752290, 19.7281021890256342332879739802, 20.68616336074606880494853912481, 21.91393312383191510749315179413, 22.63556000188671269716031006208, 23.45222781450721613535598478301, 24.04752491668103125380909386814, 25.4766780646140932619354647762, 26.312808609629194403598523699405, 26.607004973984849105407403217962