L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + 1.24i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.623 − 1.08i)10-s + (1.41 − 0.816i)11-s + (3.60 + 0.117i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−2.78 + 4.81i)17-s + (−4.23 − 2.44i)19-s + (1.08 + 0.623i)20-s + (−0.816 + 1.41i)22-s + (2.83 + 4.91i)23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.557i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.197 − 0.341i)10-s + (0.426 − 0.246i)11-s + (0.999 + 0.0325i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.674 + 1.16i)17-s + (−0.972 − 0.561i)19-s + (0.241 + 0.139i)20-s + (−0.174 + 0.301i)22-s + (0.591 + 1.02i)23-s + ⋯ |
Λ(s)=(=(1638s/2ΓC(s)L(s)(0.0453−0.998i)Λ(2−s)
Λ(s)=(=(1638s/2ΓC(s+1/2)L(s)(0.0453−0.998i)Λ(1−s)
Degree: |
2 |
Conductor: |
1638
= 2⋅32⋅7⋅13
|
Sign: |
0.0453−0.998i
|
Analytic conductor: |
13.0794 |
Root analytic conductor: |
3.61655 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1638(127,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1638, ( :1/2), 0.0453−0.998i)
|
Particular Values
L(1) |
≈ |
1.274991996 |
L(21) |
≈ |
1.274991996 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.866−0.5i)T |
| 3 | 1 |
| 7 | 1+(−0.866−0.5i)T |
| 13 | 1+(−3.60−0.117i)T |
good | 5 | 1−1.24iT−5T2 |
| 11 | 1+(−1.41+0.816i)T+(5.5−9.52i)T2 |
| 17 | 1+(2.78−4.81i)T+(−8.5−14.7i)T2 |
| 19 | 1+(4.23+2.44i)T+(9.5+16.4i)T2 |
| 23 | 1+(−2.83−4.91i)T+(−11.5+19.9i)T2 |
| 29 | 1+(1.59+2.77i)T+(−14.5+25.1i)T2 |
| 31 | 1−3.53iT−31T2 |
| 37 | 1+(−6.12+3.53i)T+(18.5−32.0i)T2 |
| 41 | 1+(−3.46+1.99i)T+(20.5−35.5i)T2 |
| 43 | 1+(−2.69+4.66i)T+(−21.5−37.2i)T2 |
| 47 | 1−11.4iT−47T2 |
| 53 | 1+4.82T+53T2 |
| 59 | 1+(4.48+2.58i)T+(29.5+51.0i)T2 |
| 61 | 1+(2.97−5.14i)T+(−30.5−52.8i)T2 |
| 67 | 1+(5.48−3.16i)T+(33.5−58.0i)T2 |
| 71 | 1+(−11.3−6.55i)T+(35.5+61.4i)T2 |
| 73 | 1+6.62iT−73T2 |
| 79 | 1+8.21T+79T2 |
| 83 | 1−11.1iT−83T2 |
| 89 | 1+(−2.16+1.24i)T+(44.5−77.0i)T2 |
| 97 | 1+(−9.02−5.20i)T+(48.5+84.0i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−9.265884128020271733089793653463, −8.850763816037150720322655432765, −8.062817365562773409663291278456, −7.20492852639242843313431630023, −6.33593770497694945790199336616, −5.89729264028420717674836552248, −4.60386170775407703592111064343, −3.63024019558302703076740925005, −2.39903115219632132371289198084, −1.23540456400531634060342546835,
0.68648295231166232757845933070, 1.80557920110899329390622678186, 2.98361541753427380116195455624, 4.20082061459965237542413678210, 4.82329642838064957510253385212, 6.13619842343519846845708320135, 6.84283370390457624343750276663, 7.80653791941733618847869241175, 8.616399129344277415818045898019, 9.046534446176418942107006654434