L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)11-s + 12-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + 17-s + ⋯ |
L(s) = 1 | + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + (−0.222 − 0.974i)5-s + (−0.222 + 0.974i)6-s + (−0.900 − 0.433i)7-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)11-s + 12-s + (0.623 − 0.781i)13-s + (−0.222 + 0.974i)14-s + (−0.222 + 0.974i)15-s + (0.623 − 0.781i)16-s + 17-s + ⋯ |
Λ(s)=(=(29s/2ΓR(s)L(s)(−0.833−0.552i)Λ(1−s)
Λ(s)=(=(29s/2ΓR(s)L(s)(−0.833−0.552i)Λ(1−s)
Degree: |
1 |
Conductor: |
29
|
Sign: |
−0.833−0.552i
|
Analytic conductor: |
0.134675 |
Root analytic conductor: |
0.134675 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ29(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 29, (0: ), −0.833−0.552i)
|
Particular Values
L(21) |
≈ |
0.1252539855−0.4153005135i |
L(21) |
≈ |
0.1252539855−0.4153005135i |
L(1) |
≈ |
0.4060286231−0.4387473455i |
L(1) |
≈ |
0.4060286231−0.4387473455i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 29 | 1 |
good | 2 | 1+(−0.222−0.974i)T |
| 3 | 1+(−0.900−0.433i)T |
| 5 | 1+(−0.222−0.974i)T |
| 7 | 1+(−0.900−0.433i)T |
| 11 | 1+(0.623−0.781i)T |
| 13 | 1+(0.623−0.781i)T |
| 17 | 1+T |
| 19 | 1+(−0.900+0.433i)T |
| 23 | 1+(−0.222+0.974i)T |
| 31 | 1+(−0.222−0.974i)T |
| 37 | 1+(0.623+0.781i)T |
| 41 | 1+T |
| 43 | 1+(−0.222+0.974i)T |
| 47 | 1+(0.623−0.781i)T |
| 53 | 1+(−0.222−0.974i)T |
| 59 | 1+T |
| 61 | 1+(−0.900−0.433i)T |
| 67 | 1+(0.623+0.781i)T |
| 71 | 1+(0.623−0.781i)T |
| 73 | 1+(−0.222+0.974i)T |
| 79 | 1+(0.623+0.781i)T |
| 83 | 1+(−0.900+0.433i)T |
| 89 | 1+(−0.222−0.974i)T |
| 97 | 1+(−0.900+0.433i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−38.099194941985007198342838717, −36.08608405874741486374479895124, −35.002027431403321018488499625414, −34.19187594451787179787235486982, −33.155331828851572117256530086165, −32.046577545962595831616672116221, −30.37317107927239225701839162902, −28.63859256571346564009026616026, −27.65166844714059773299516858789, −26.36128886257882442925785325811, −25.407223089819509606002718526283, −23.50134608875226177624719111419, −22.76107567693998433608008161269, −21.74189713712751517079452276206, −19.15677544379112439448877309146, −18.11746922850488862886385442705, −16.71966022143983979342183067697, −15.63383290726376302209832182125, −14.47081709930705995664168065457, −12.39192319389359990463903614070, −10.55565615586021738439767506672, −9.27687897191694663018669806762, −6.95260585981191078176305950675, −6.076938513657141513085772390860, −4.08081536101879485562656864049,
1.00015516298170237827229722153, 3.84734117535889590769822772057, 5.76138533550202578842369012, 8.06265765543259523867381909386, 9.81319872187136677468355662618, 11.30354467973762113377730018629, 12.564350442222234121164453558404, 13.41939603830550997037822142997, 16.35821936653212835501551419773, 17.184078559834183717762828146357, 18.81356811761612836733901966176, 19.829941058161542155152524800035, 21.31904373946677078635544504099, 22.71765027143154057888121936589, 23.661221899116944224779909940291, 25.40734523332958955083670525640, 27.38146516208638100938692513765, 28.0898412100639471747909975924, 29.386163400171109665340805557718, 29.99144329817097919310611084697, 31.796202841561475843714425398355, 32.78775369590765265901536744270, 34.91493620124064258561448220371, 35.6462944652756531689059482402, 36.57927284035358527020775944502