L(s) = 1 | + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.104 − 0.994i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.669 + 0.743i)26-s + (0.669 + 0.743i)29-s + (0.104 + 0.994i)31-s + (−0.5 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.913 + 0.406i)2-s + (0.669 + 0.743i)4-s + (−0.809 − 0.587i)5-s + (0.309 + 0.951i)8-s + (−0.5 − 0.866i)10-s + (0.913 + 0.406i)13-s + (−0.104 + 0.994i)16-s + (0.104 − 0.994i)17-s + (−0.978 − 0.207i)19-s + (−0.104 − 0.994i)20-s − 23-s + (0.309 + 0.951i)25-s + (0.669 + 0.743i)26-s + (0.669 + 0.743i)29-s + (0.104 + 0.994i)31-s + (−0.5 + 0.866i)32-s + ⋯ |
Λ(s)=(=(693s/2ΓR(s+1)L(s)(−0.675+0.737i)Λ(1−s)
Λ(s)=(=(693s/2ΓR(s+1)L(s)(−0.675+0.737i)Λ(1−s)
Degree: |
1 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
−0.675+0.737i
|
Analytic conductor: |
74.4731 |
Root analytic conductor: |
74.4731 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(248,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 693, (1: ), −0.675+0.737i)
|
Particular Values
L(21) |
≈ |
0.8838710812+2.009794108i |
L(21) |
≈ |
0.8838710812+2.009794108i |
L(1) |
≈ |
1.393632717+0.5229194884i |
L(1) |
≈ |
1.393632717+0.5229194884i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1 |
| 11 | 1 |
good | 2 | 1+(0.913+0.406i)T |
| 5 | 1+(−0.809−0.587i)T |
| 13 | 1+(0.913+0.406i)T |
| 17 | 1+(0.104−0.994i)T |
| 19 | 1+(−0.978−0.207i)T |
| 23 | 1−T |
| 29 | 1+(0.669+0.743i)T |
| 31 | 1+(0.104+0.994i)T |
| 37 | 1+(0.669+0.743i)T |
| 41 | 1+(−0.669+0.743i)T |
| 43 | 1+(0.5−0.866i)T |
| 47 | 1+(0.669−0.743i)T |
| 53 | 1+(−0.913−0.406i)T |
| 59 | 1+(0.669+0.743i)T |
| 61 | 1+(−0.104+0.994i)T |
| 67 | 1+(−0.5+0.866i)T |
| 71 | 1+(0.809+0.587i)T |
| 73 | 1+(−0.978+0.207i)T |
| 79 | 1+(−0.913−0.406i)T |
| 83 | 1+(−0.913+0.406i)T |
| 89 | 1+(−0.5+0.866i)T |
| 97 | 1+(0.104+0.994i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.24681208376395303897889187767, −21.44379353755693073429558574476, −20.62365580205257769462257670448, −19.78528006386336209300212162758, −19.10285160826972197191717484643, −18.42304466007895512102437681979, −17.21207155680809785126313281902, −15.99486716086458220727742324738, −15.47918588168277492639843780110, −14.68443711277443798466200807136, −13.93760027127476474834795392593, −12.903289529314136338393349209120, −12.25744454487306710486236307031, −11.25094114203464801644740089629, −10.73272030908713708989438128667, −9.86949948281457647021396561699, −8.37871983855929298826857870646, −7.61162688191896797709796640255, −6.33036950525714934782002524464, −5.93832651096578586246564968963, −4.36594196071017517082018252203, −3.88890829637594813719086292721, −2.88193438606886888686331976688, −1.79724184268279220453116676890, −0.36405123707584091155287281314,
1.27646905957850666063669520973, 2.69255716050029319478804993280, 3.77287974703499636017960002233, 4.475967908251458096558231691783, 5.35252529627822477064276010017, 6.45944576159616969762799889818, 7.24508938730460414571616142508, 8.310000243948046999515415392487, 8.83522211000452720992665061119, 10.3906853363738100308541352754, 11.45470059202247892800948016288, 11.97549592808137965466687784917, 12.87255283535570998573073841918, 13.6556016157116012948156469943, 14.49077616991378020896237159196, 15.47048627912846480972674381318, 16.09218625026088583756381177410, 16.63380339489836929627208172817, 17.71234097758715750062650824624, 18.75109880283773294650643429849, 19.835145427753134682335887342800, 20.42171757284220431378625384543, 21.2177310631016528380813492847, 22.02132892400304603571421830842, 22.99581762836283901980203159297