L(s) = 1 | + (2.04 − 1.48i)2-s + (0.309 + 0.951i)3-s + (1.35 − 4.15i)4-s + (1.91 + 1.39i)5-s + (2.04 + 1.48i)6-s + (−0.244 + 0.753i)7-s + (−1.85 − 5.69i)8-s + (−0.809 + 0.587i)9-s + 5.98·10-s + 4.37·12-s + (−3.32 + 2.41i)13-s + (0.618 + 1.90i)14-s + (−0.733 + 2.25i)15-s + (−5.15 − 3.74i)16-s + (−4.84 − 3.51i)17-s + (−0.780 + 2.40i)18-s + ⋯ |
L(s) = 1 | + (1.44 − 1.04i)2-s + (0.178 + 0.549i)3-s + (0.675 − 2.07i)4-s + (0.858 + 0.623i)5-s + (0.833 + 0.605i)6-s + (−0.0925 + 0.284i)7-s + (−0.654 − 2.01i)8-s + (−0.269 + 0.195i)9-s + 1.89·10-s + 1.26·12-s + (−0.921 + 0.669i)13-s + (0.165 + 0.508i)14-s + (−0.189 + 0.582i)15-s + (−1.28 − 0.936i)16-s + (−1.17 − 0.853i)17-s + (−0.183 + 0.565i)18-s + ⋯ |
Λ(s)=(=(363s/2ΓC(s)L(s)(0.659+0.751i)Λ(2−s)
Λ(s)=(=(363s/2ΓC(s+1/2)L(s)(0.659+0.751i)Λ(1−s)
Degree: |
2 |
Conductor: |
363
= 3⋅112
|
Sign: |
0.659+0.751i
|
Analytic conductor: |
2.89856 |
Root analytic conductor: |
1.70251 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ363(124,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 363, ( :1/2), 0.659+0.751i)
|
Particular Values
L(1) |
≈ |
2.84968−1.29114i |
L(21) |
≈ |
2.84968−1.29114i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.309−0.951i)T |
| 11 | 1 |
good | 2 | 1+(−2.04+1.48i)T+(0.618−1.90i)T2 |
| 5 | 1+(−1.91−1.39i)T+(1.54+4.75i)T2 |
| 7 | 1+(0.244−0.753i)T+(−5.66−4.11i)T2 |
| 13 | 1+(3.32−2.41i)T+(4.01−12.3i)T2 |
| 17 | 1+(4.84+3.51i)T+(5.25+16.1i)T2 |
| 19 | 1+(1.31+4.04i)T+(−15.3+11.1i)T2 |
| 23 | 1−2T+23T2 |
| 29 | 1+(−0.780+2.40i)T+(−23.4−17.0i)T2 |
| 31 | 1+(5.96−4.33i)T+(9.57−29.4i)T2 |
| 37 | 1+(−1.54+4.75i)T+(−29.9−21.7i)T2 |
| 41 | 1+(−1.75−5.41i)T+(−33.1+24.0i)T2 |
| 43 | 1−6.63T+43T2 |
| 47 | 1+(0.387+1.19i)T+(−38.0+27.6i)T2 |
| 53 | 1+(10.6−7.70i)T+(16.3−50.4i)T2 |
| 59 | 1+(1.85−5.70i)T+(−47.7−34.6i)T2 |
| 61 | 1+(2.16+1.57i)T+(18.8+58.0i)T2 |
| 67 | 1−16.1T+67T2 |
| 71 | 1+(0.602+0.437i)T+(21.9+67.5i)T2 |
| 73 | 1+(2.29−7.06i)T+(−59.0−42.9i)T2 |
| 79 | 1+(−4.72+3.43i)T+(24.4−75.1i)T2 |
| 83 | 1+(−6.88−5.00i)T+(25.6+78.9i)T2 |
| 89 | 1+6.37T+89T2 |
| 97 | 1+(−10.1+7.34i)T+(29.9−92.2i)T2 |
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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
−11.18767086752607706112717163110, −10.80339735622160562169578449168, −9.676402940056464003672878268581, −9.142831784581604241495592782077, −7.05433935748048453147273445344, −6.11467717295028886089627366000, −5.03959251819577741238894872750, −4.27922432912401501028706396725, −2.80196072639219902061330052493, −2.25003971314852961356507237130,
2.17223613167512368719926516358, 3.68043327897373809642504144276, 4.88395399425372210510049681943, 5.73585399959510680874266753330, 6.53483924294312865866600443418, 7.50090382129395780465298103434, 8.363472265141825207909613869262, 9.507086387558719661938609624344, 10.84771112839467234918442898736, 12.19288765028738570115195504986