L(s) = 1 | + (−1.71 − 0.272i)3-s + (−0.119 + 0.207i)5-s + (0.5 + 0.866i)7-s + (2.85 + 0.931i)9-s + (2.56 + 4.43i)11-s + (2.44 − 4.23i)13-s + (0.260 − 0.321i)15-s + 3.70·17-s + 3.66·19-s + (−0.619 − 1.61i)21-s + (−3.71 + 6.42i)23-s + (2.47 + 4.28i)25-s + (−4.62 − 2.36i)27-s + (−1.73 − 3.00i)29-s + (0.358 − 0.621i)31-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.157i)3-s + (−0.0534 + 0.0926i)5-s + (0.188 + 0.327i)7-s + (0.950 + 0.310i)9-s + (0.772 + 1.33i)11-s + (0.677 − 1.17i)13-s + (0.0673 − 0.0830i)15-s + 0.898·17-s + 0.839·19-s + (−0.135 − 0.352i)21-s + (−0.773 + 1.34i)23-s + (0.494 + 0.856i)25-s + (−0.890 − 0.455i)27-s + (−0.321 − 0.557i)29-s + (0.0644 − 0.111i)31-s + ⋯ |
Λ(s)=(=(252s/2ΓC(s)L(s)(0.927−0.373i)Λ(2−s)
Λ(s)=(=(252s/2ΓC(s+1/2)L(s)(0.927−0.373i)Λ(1−s)
Degree: |
2 |
Conductor: |
252
= 22⋅32⋅7
|
Sign: |
0.927−0.373i
|
Analytic conductor: |
2.01223 |
Root analytic conductor: |
1.41853 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ252(85,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 252, ( :1/2), 0.927−0.373i)
|
Particular Values
L(1) |
≈ |
0.987593+0.191240i |
L(21) |
≈ |
0.987593+0.191240i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(1.71+0.272i)T |
| 7 | 1+(−0.5−0.866i)T |
good | 5 | 1+(0.119−0.207i)T+(−2.5−4.33i)T2 |
| 11 | 1+(−2.56−4.43i)T+(−5.5+9.52i)T2 |
| 13 | 1+(−2.44+4.23i)T+(−6.5−11.2i)T2 |
| 17 | 1−3.70T+17T2 |
| 19 | 1−3.66T+19T2 |
| 23 | 1+(3.71−6.42i)T+(−11.5−19.9i)T2 |
| 29 | 1+(1.73+3.00i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−0.358+0.621i)T+(−15.5−26.8i)T2 |
| 37 | 1−4.60T+37T2 |
| 41 | 1+(2.80−4.85i)T+(−20.5−35.5i)T2 |
| 43 | 1+(6.24+10.8i)T+(−21.5+37.2i)T2 |
| 47 | 1+(2.16+3.75i)T+(−23.5+40.7i)T2 |
| 53 | 1+0.942T+53T2 |
| 59 | 1+(3.78−6.56i)T+(−29.5−51.0i)T2 |
| 61 | 1+(2.75+4.77i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−0.330+0.571i)T+(−33.5−58.0i)T2 |
| 71 | 1−13.7T+71T2 |
| 73 | 1+3.66T+73T2 |
| 79 | 1+(−3.11−5.39i)T+(−39.5+68.4i)T2 |
| 83 | 1+(4.85+8.40i)T+(−41.5+71.8i)T2 |
| 89 | 1+7.48T+89T2 |
| 97 | 1+(−8.57−14.8i)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
−11.97799351870408510048550690269, −11.37693140805465257400742687260, −10.17715003106063684072318641559, −9.527149679089405759578863469670, −7.930476536145668280255029103203, −7.12971908713605981445671675979, −5.88775049970322527640907444623, −5.09276305379219344796297682095, −3.63719118170593401263908072317, −1.49537085468508805979606334047,
1.13148452339232045201304986063, 3.60962090874277268982833436381, 4.69991105010304886346018262146, 6.02837776730201479444681698013, 6.67220598287954726968658860231, 8.090643612147532324380121976392, 9.178984024196770397496498408289, 10.25084022203673387344958845302, 11.22211114282473288925936106151, 11.74350293369536384169631102504