L(s) = 1 | + (1.40 − 0.116i)2-s + (0.312 + 1.16i)3-s + (1.97 − 0.327i)4-s + (0.262 − 0.980i)5-s + (0.575 + 1.60i)6-s + (2.74 − 0.690i)8-s + (1.33 − 0.772i)9-s + (0.256 − 1.41i)10-s + (−2.36 + 0.635i)11-s + (0.997 + 2.19i)12-s + (2.65 − 2.65i)13-s + 1.22·15-s + (3.78 − 1.29i)16-s + (0.509 − 0.881i)17-s + (1.79 − 1.24i)18-s + (−0.0936 − 0.0250i)19-s + ⋯ |
L(s) = 1 | + (0.996 − 0.0820i)2-s + (0.180 + 0.672i)3-s + (0.986 − 0.163i)4-s + (0.117 − 0.438i)5-s + (0.234 + 0.655i)6-s + (0.969 − 0.243i)8-s + (0.445 − 0.257i)9-s + (0.0811 − 0.446i)10-s + (−0.714 + 0.191i)11-s + (0.287 + 0.634i)12-s + (0.737 − 0.737i)13-s + 0.316·15-s + (0.946 − 0.322i)16-s + (0.123 − 0.213i)17-s + (0.423 − 0.293i)18-s + (−0.0214 − 0.00575i)19-s + ⋯ |
Λ(s)=(=(784s/2ΓC(s)L(s)(0.998−0.0626i)Λ(2−s)
Λ(s)=(=(784s/2ΓC(s+1/2)L(s)(0.998−0.0626i)Λ(1−s)
Degree: |
2 |
Conductor: |
784
= 24⋅72
|
Sign: |
0.998−0.0626i
|
Analytic conductor: |
6.26027 |
Root analytic conductor: |
2.50205 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ784(165,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 784, ( :1/2), 0.998−0.0626i)
|
Particular Values
L(1) |
≈ |
3.27190+0.102553i |
L(21) |
≈ |
3.27190+0.102553i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.40+0.116i)T |
| 7 | 1 |
good | 3 | 1+(−0.312−1.16i)T+(−2.59+1.5i)T2 |
| 5 | 1+(−0.262+0.980i)T+(−4.33−2.5i)T2 |
| 11 | 1+(2.36−0.635i)T+(9.52−5.5i)T2 |
| 13 | 1+(−2.65+2.65i)T−13iT2 |
| 17 | 1+(−0.509+0.881i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.0936+0.0250i)T+(16.4+9.5i)T2 |
| 23 | 1+(1.67−0.965i)T+(11.5−19.9i)T2 |
| 29 | 1+(5.05−5.05i)T−29iT2 |
| 31 | 1+(4.28−7.41i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−2.06+7.71i)T+(−32.0−18.5i)T2 |
| 41 | 1−8.51iT−41T2 |
| 43 | 1+(4.47+4.47i)T+43iT2 |
| 47 | 1+(6.02+10.4i)T+(−23.5+40.7i)T2 |
| 53 | 1+(1.42−0.381i)T+(45.8−26.5i)T2 |
| 59 | 1+(6.96−1.86i)T+(51.0−29.5i)T2 |
| 61 | 1+(4.42+1.18i)T+(52.8+30.5i)T2 |
| 67 | 1+(−0.907−3.38i)T+(−58.0+33.5i)T2 |
| 71 | 1−5.43iT−71T2 |
| 73 | 1+(−7.34−4.23i)T+(36.5+63.2i)T2 |
| 79 | 1+(0.433+0.751i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−5.44+5.44i)T−83iT2 |
| 89 | 1+(−3.93+2.26i)T+(44.5−77.0i)T2 |
| 97 | 1+16.4T+97T2 |
show more | |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.51748010851719834336726701919, −9.621111482334147332873115312214, −8.660912380437285242423857200410, −7.57553892870616608077747711759, −6.70125138160333286544778900634, −5.45305914644131868250873713726, −4.99353309687155995309557245783, −3.82741163945073909784510100178, −3.10941964066064380804826334207, −1.52501506943628373253936706502,
1.69126367009939234938540605752, 2.65603157807735266491138065977, 3.86064863665308237556786035273, 4.84103665079545321833850202881, 6.07444464292736855794709912846, 6.58574069203249906011518959998, 7.63613736830069312128542268759, 8.113375511236834123374377468817, 9.551206655716848005183499208910, 10.61171754981044908946722477686