L(s) = 1 | + (−0.863 − 0.627i)2-s + (−0.266 − 0.819i)4-s + (−0.993 + 0.721i)5-s + (−0.473 − 1.45i)7-s + (−0.943 + 2.90i)8-s + 1.30·10-s + (−3.00 − 1.40i)11-s + (−1.89 − 1.37i)13-s + (−0.505 + 1.55i)14-s + (1.24 − 0.900i)16-s + (−2.87 + 2.08i)17-s + (−0.884 + 2.72i)19-s + (0.856 + 0.622i)20-s + (1.70 + 3.09i)22-s − 2.29·23-s + ⋯ |
L(s) = 1 | + (−0.610 − 0.443i)2-s + (−0.133 − 0.409i)4-s + (−0.444 + 0.322i)5-s + (−0.179 − 0.550i)7-s + (−0.333 + 1.02i)8-s + 0.414·10-s + (−0.905 − 0.424i)11-s + (−0.525 − 0.381i)13-s + (−0.135 + 0.415i)14-s + (0.310 − 0.225i)16-s + (−0.696 + 0.506i)17-s + (−0.202 + 0.624i)19-s + (0.191 + 0.139i)20-s + (0.364 + 0.660i)22-s − 0.478·23-s + ⋯ |
Λ(s)=(=(297s/2ΓC(s)L(s)(−0.915−0.402i)Λ(2−s)
Λ(s)=(=(297s/2ΓC(s+1/2)L(s)(−0.915−0.402i)Λ(1−s)
Degree: |
2 |
Conductor: |
297
= 33⋅11
|
Sign: |
−0.915−0.402i
|
Analytic conductor: |
2.37155 |
Root analytic conductor: |
1.53998 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ297(136,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 297, ( :1/2), −0.915−0.402i)
|
Particular Values
L(1) |
≈ |
0.0343438+0.163395i |
L(21) |
≈ |
0.0343438+0.163395i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 11 | 1+(3.00+1.40i)T |
good | 2 | 1+(0.863+0.627i)T+(0.618+1.90i)T2 |
| 5 | 1+(0.993−0.721i)T+(1.54−4.75i)T2 |
| 7 | 1+(0.473+1.45i)T+(−5.66+4.11i)T2 |
| 13 | 1+(1.89+1.37i)T+(4.01+12.3i)T2 |
| 17 | 1+(2.87−2.08i)T+(5.25−16.1i)T2 |
| 19 | 1+(0.884−2.72i)T+(−15.3−11.1i)T2 |
| 23 | 1+2.29T+23T2 |
| 29 | 1+(2.56+7.88i)T+(−23.4+17.0i)T2 |
| 31 | 1+(3.91+2.84i)T+(9.57+29.4i)T2 |
| 37 | 1+(−1.24−3.82i)T+(−29.9+21.7i)T2 |
| 41 | 1+(1.49−4.60i)T+(−33.1−24.0i)T2 |
| 43 | 1+7.52T+43T2 |
| 47 | 1+(−2.82+8.70i)T+(−38.0−27.6i)T2 |
| 53 | 1+(−5.00−3.63i)T+(16.3+50.4i)T2 |
| 59 | 1+(3.36+10.3i)T+(−47.7+34.6i)T2 |
| 61 | 1+(−0.496+0.360i)T+(18.8−58.0i)T2 |
| 67 | 1−8.65T+67T2 |
| 71 | 1+(−12.6+9.18i)T+(21.9−67.5i)T2 |
| 73 | 1+(4.68+14.4i)T+(−59.0+42.9i)T2 |
| 79 | 1+(−1.57−1.14i)T+(24.4+75.1i)T2 |
| 83 | 1+(4.31−3.13i)T+(25.6−78.9i)T2 |
| 89 | 1+5.84T+89T2 |
| 97 | 1+(−6.88−4.99i)T+(29.9+92.2i)T2 |
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
−11.01785488061966085198825763108, −10.30992993084841102244650156866, −9.612288591408481281068213843944, −8.337049816485317838465616202152, −7.65744000420305026361764695214, −6.25973518668195910053589317502, −5.15532487909200089231144101595, −3.70880228987770723634052324784, −2.17848557313295743388748027779, −0.14107557477467729452769360210,
2.58842987220199569296241443061, 4.12326549315683941562656031929, 5.27978323363656945273410871358, 6.81365534189727916096868867584, 7.51223564356333485912094512236, 8.574328336225043351652546671242, 9.161330873096570337317671737964, 10.19287018484410279695984229918, 11.41536759079585832460009151459, 12.46135186472887520806645283054