L(s) = 1 | − i·2-s − 4-s + (−2.17 + 0.5i)5-s − 4.35i·7-s + i·8-s + (0.5 + 2.17i)10-s − 4.35·11-s − 4.35·14-s + 16-s − 4i·17-s − 6·19-s + (2.17 − 0.5i)20-s + 4.35i·22-s − 2i·23-s + (4.50 − 2.17i)25-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.5·4-s + (−0.974 + 0.223i)5-s − 1.64i·7-s + 0.353i·8-s + (0.158 + 0.689i)10-s − 1.31·11-s − 1.16·14-s + 0.250·16-s − 0.970i·17-s − 1.37·19-s + (0.487 − 0.111i)20-s + 0.929i·22-s − 0.417i·23-s + (0.900 − 0.435i)25-s + ⋯ |
Λ(s)=(=(270s/2ΓC(s)L(s)(−0.974+0.223i)Λ(2−s)
Λ(s)=(=(270s/2ΓC(s+1/2)L(s)(−0.974+0.223i)Λ(1−s)
Degree: |
2 |
Conductor: |
270
= 2⋅33⋅5
|
Sign: |
−0.974+0.223i
|
Analytic conductor: |
2.15596 |
Root analytic conductor: |
1.46831 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ270(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 270, ( :1/2), −0.974+0.223i)
|
Particular Values
L(1) |
≈ |
0.0701231−0.619259i |
L(21) |
≈ |
0.0701231−0.619259i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+iT |
| 3 | 1 |
| 5 | 1+(2.17−0.5i)T |
good | 7 | 1+4.35iT−7T2 |
| 11 | 1+4.35T+11T2 |
| 13 | 1−13T2 |
| 17 | 1+4iT−17T2 |
| 19 | 1+6T+19T2 |
| 23 | 1+2iT−23T2 |
| 29 | 1+29T2 |
| 31 | 1−7T+31T2 |
| 37 | 1−8.71iT−37T2 |
| 41 | 1−8.71T+41T2 |
| 43 | 1+8.71iT−43T2 |
| 47 | 1+2iT−47T2 |
| 53 | 1+3iT−53T2 |
| 59 | 1−8.71T+59T2 |
| 61 | 1+4T+61T2 |
| 67 | 1+8.71iT−67T2 |
| 71 | 1+71T2 |
| 73 | 1−4.35iT−73T2 |
| 79 | 1+79T2 |
| 83 | 1+5iT−83T2 |
| 89 | 1+8.71T+89T2 |
| 97 | 1−4.35iT−97T2 |
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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
−11.29258745136216931977106285123, −10.60384238929494622561451855758, −10.03597991959064327200141034255, −8.455040859108733109926187603313, −7.68264471490111978070905826614, −6.74297340677452860664192877079, −4.83113162903713253478382194672, −4.04094593557720861627866626263, −2.77550267018518104598069753251, −0.46931538822514545683082698665,
2.61509841044716268569206278134, 4.25958593489423691737548826188, 5.39705184573516012405240074521, 6.27720123821195438392519419345, 7.74882062897673679131828797956, 8.351675457227233897594458238514, 9.124065748021794419120202126013, 10.49092777858386068241926224544, 11.51685776263620046715829016576, 12.67467644605015058570294106258