L(s) = 1 | + (0.5 − 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.681 + 1.18i)5-s + (0.866 − 0.499i)6-s − 0.440i·7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (0.681 + 1.18i)10-s + (2.75 + 1.85i)11-s − 0.999i·12-s + (2.12 + 3.67i)13-s + (−0.381 − 0.220i)14-s + (−1.18 + 0.681i)15-s + (−0.5 + 0.866i)16-s + (−1.61 − 0.930i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.304 + 0.528i)5-s + (0.353 − 0.204i)6-s − 0.166i·7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.215 + 0.373i)10-s + (0.829 + 0.557i)11-s − 0.288i·12-s + (0.588 + 1.02i)13-s + (−0.101 − 0.0588i)14-s + (−0.304 + 0.176i)15-s + (−0.125 + 0.216i)16-s + (−0.390 − 0.225i)17-s + ⋯ |
Λ(s)=(=(1254s/2ΓC(s)L(s)(0.889−0.456i)Λ(2−s)
Λ(s)=(=(1254s/2ΓC(s+1/2)L(s)(0.889−0.456i)Λ(1−s)
Degree: |
2 |
Conductor: |
1254
= 2⋅3⋅11⋅19
|
Sign: |
0.889−0.456i
|
Analytic conductor: |
10.0132 |
Root analytic conductor: |
3.16437 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1254(901,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1254, ( :1/2), 0.889−0.456i)
|
Particular Values
L(1) |
≈ |
2.149740373 |
L(21) |
≈ |
2.149740373 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.5+0.866i)T |
| 3 | 1+(−0.866−0.5i)T |
| 11 | 1+(−2.75−1.85i)T |
| 19 | 1+(4.34+0.344i)T |
good | 5 | 1+(0.681−1.18i)T+(−2.5−4.33i)T2 |
| 7 | 1+0.440iT−7T2 |
| 13 | 1+(−2.12−3.67i)T+(−6.5+11.2i)T2 |
| 17 | 1+(1.61+0.930i)T+(8.5+14.7i)T2 |
| 23 | 1+(−3.53−6.12i)T+(−11.5+19.9i)T2 |
| 29 | 1+(2.67+4.63i)T+(−14.5+25.1i)T2 |
| 31 | 1+0.0632iT−31T2 |
| 37 | 1−7.40iT−37T2 |
| 41 | 1+(−0.0722+0.125i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−7.46−4.31i)T+(21.5+37.2i)T2 |
| 47 | 1+(6.08+10.5i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−2.86+1.65i)T+(26.5−45.8i)T2 |
| 59 | 1+(−11.3−6.57i)T+(29.5+51.0i)T2 |
| 61 | 1+(−12.1+7.04i)T+(30.5−52.8i)T2 |
| 67 | 1+(7.77−4.48i)T+(33.5−58.0i)T2 |
| 71 | 1+(−6.03−3.48i)T+(35.5+61.4i)T2 |
| 73 | 1+(0.848+0.489i)T+(36.5+63.2i)T2 |
| 79 | 1+(−0.259+0.449i)T+(−39.5−68.4i)T2 |
| 83 | 1−0.315iT−83T2 |
| 89 | 1+(8.87−5.12i)T+(44.5−77.0i)T2 |
| 97 | 1+(0.113+0.0657i)T+(48.5+84.0i)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
−9.708682097117252926957296074556, −9.152781696629962285566129357612, −8.338392852145221087625096111718, −7.10122340471865327738987287865, −6.62181339708858381313712170411, −5.33369090467911615678039514981, −4.15660027351611044379613009533, −3.81108087406114797867289046823, −2.59590655151706720214309533632, −1.51976944621083041946145969967,
0.823608547775802055471086840518, 2.50666443373098551183830154313, 3.67217617188036042638541473922, 4.39171229354510910606476767381, 5.53882616685595990829912411457, 6.34625048492379781132873220237, 7.11008460298782158895421130783, 8.180178806307375713943482759506, 8.652474746068101749606714161172, 9.143488329865804608736310537511