L(s) = 1 | + (0.997 + 0.0747i)3-s + (−0.563 + 0.826i)5-s + (0.988 + 0.149i)9-s + (−0.149 − 0.988i)11-s + (0.781 − 0.623i)13-s + (−0.623 + 0.781i)15-s + (−0.955 − 0.294i)17-s + (0.866 − 0.5i)19-s + (0.955 − 0.294i)23-s + (−0.365 − 0.930i)25-s + (0.974 + 0.222i)27-s + (0.974 − 0.222i)29-s + (−0.5 + 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.680 + 0.733i)37-s + ⋯ |
L(s) = 1 | + (0.997 + 0.0747i)3-s + (−0.563 + 0.826i)5-s + (0.988 + 0.149i)9-s + (−0.149 − 0.988i)11-s + (0.781 − 0.623i)13-s + (−0.623 + 0.781i)15-s + (−0.955 − 0.294i)17-s + (0.866 − 0.5i)19-s + (0.955 − 0.294i)23-s + (−0.365 − 0.930i)25-s + (0.974 + 0.222i)27-s + (0.974 − 0.222i)29-s + (−0.5 + 0.866i)31-s + (−0.0747 − 0.997i)33-s + (0.680 + 0.733i)37-s + ⋯ |
Λ(s)=(=(784s/2ΓR(s)L(s)(0.999+0.00801i)Λ(1−s)
Λ(s)=(=(784s/2ΓR(s)L(s)(0.999+0.00801i)Λ(1−s)
Degree: |
1 |
Conductor: |
784
= 24⋅72
|
Sign: |
0.999+0.00801i
|
Analytic conductor: |
3.64088 |
Root analytic conductor: |
3.64088 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ784(355,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 784, (0: ), 0.999+0.00801i)
|
Particular Values
L(21) |
≈ |
1.999883121+0.008013841519i |
L(21) |
≈ |
1.999883121+0.008013841519i |
L(1) |
≈ |
1.435675967+0.06052251388i |
L(1) |
≈ |
1.435675967+0.06052251388i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
good | 3 | 1+(0.997+0.0747i)T |
| 5 | 1+(−0.563+0.826i)T |
| 11 | 1+(−0.149−0.988i)T |
| 13 | 1+(0.781−0.623i)T |
| 17 | 1+(−0.955−0.294i)T |
| 19 | 1+(0.866−0.5i)T |
| 23 | 1+(0.955−0.294i)T |
| 29 | 1+(0.974−0.222i)T |
| 31 | 1+(−0.5+0.866i)T |
| 37 | 1+(0.680+0.733i)T |
| 41 | 1+(−0.900+0.433i)T |
| 43 | 1+(0.433−0.900i)T |
| 47 | 1+(0.365−0.930i)T |
| 53 | 1+(−0.680+0.733i)T |
| 59 | 1+(−0.563−0.826i)T |
| 61 | 1+(0.680+0.733i)T |
| 67 | 1+(0.866+0.5i)T |
| 71 | 1+(−0.222+0.974i)T |
| 73 | 1+(0.365+0.930i)T |
| 79 | 1+(0.5+0.866i)T |
| 83 | 1+(0.781+0.623i)T |
| 89 | 1+(−0.988−0.149i)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
−22.30456525669599198274779991731, −21.1481806914982702243005710910, −20.66309173080035163215996949061, −19.96816322469962278984304093103, −19.32182860955700525144542296672, −18.423139833082109373105407102866, −17.584965648568363861927692239497, −16.443763210920441685132629552598, −15.73267595793127521094261311773, −15.11669882131897744319965725658, −14.160759192415340422999077716793, −13.24409755954959924347097021348, −12.694946684329103248771115352349, −11.762741273867847889403856016620, −10.743772966196934506885558679, −9.45779076205156148658685873057, −9.08635884117351501035025996570, −8.07945132724292526852664051514, −7.43845223317535555121422266407, −6.42751665093464065617418882026, −4.95689087562723167260815076786, −4.235572003566357158647383506583, −3.39582762574909514767199027201, −2.10942518914987146509871356289, −1.20720994865030490017863231713,
1.00695091794429991258476635533, 2.62721707056593323022442341319, 3.13324512780846378660607061603, 4.000422796001193917301688331353, 5.17642206680205402590156535163, 6.542451623844132077027494019134, 7.19658894616854927541931286488, 8.284924211818377013813854913987, 8.70908215890440434047861697320, 9.88982475381355037721985289465, 10.81982463208725965215027947000, 11.38238709012518699562257977346, 12.65489386460690813958711237216, 13.65296268826921429417782853148, 14.007008013587599478666898434138, 15.20227846473482037708652073397, 15.57432358792472762336151566137, 16.359286265603694322755442693633, 17.78590481142646127683320433214, 18.51542531760716672092999378540, 19.07698475020892917478878173954, 19.99942374226741659087451254079, 20.47954136971685631775119193790, 21.65528126770677281267779842677, 22.10039663844365904301012976755