L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (0.389 + 0.224i)7-s − 0.999i·8-s + (−2.44 + 4.24i)11-s + (0.389 − 0.224i)13-s + (−0.224 − 0.389i)14-s + (−0.5 + 0.866i)16-s − 4.89i·17-s − 7.44·19-s + (4.24 − 2.44i)22-s + (−2.12 + 1.22i)23-s − 0.449·26-s + 0.449i·28-s + (1.22 − 2.12i)29-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.147 + 0.0849i)7-s − 0.353i·8-s + (−0.738 + 1.27i)11-s + (0.107 − 0.0623i)13-s + (−0.0600 − 0.104i)14-s + (−0.125 + 0.216i)16-s − 1.18i·17-s − 1.70·19-s + (0.904 − 0.522i)22-s + (−0.442 + 0.255i)23-s − 0.0881·26-s + 0.0849i·28-s + (0.227 − 0.393i)29-s + ⋯ |
Λ(s)=(=(1350s/2ΓC(s)L(s)(−0.980+0.195i)Λ(2−s)
Λ(s)=(=(1350s/2ΓC(s+1/2)L(s)(−0.980+0.195i)Λ(1−s)
Degree: |
2 |
Conductor: |
1350
= 2⋅33⋅52
|
Sign: |
−0.980+0.195i
|
Analytic conductor: |
10.7798 |
Root analytic conductor: |
3.28326 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1350(1099,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1350, ( :1/2), −0.980+0.195i)
|
Particular Values
L(1) |
≈ |
0.2130683417 |
L(21) |
≈ |
0.2130683417 |
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 |
| 5 | 1 |
good | 7 | 1+(−0.389−0.224i)T+(3.5+6.06i)T2 |
| 11 | 1+(2.44−4.24i)T+(−5.5−9.52i)T2 |
| 13 | 1+(−0.389+0.224i)T+(6.5−11.2i)T2 |
| 17 | 1+4.89iT−17T2 |
| 19 | 1+7.44T+19T2 |
| 23 | 1+(2.12−1.22i)T+(11.5−19.9i)T2 |
| 29 | 1+(−1.22+2.12i)T+(−14.5−25.1i)T2 |
| 31 | 1+(2.22+3.85i)T+(−15.5+26.8i)T2 |
| 37 | 1+11.3iT−37T2 |
| 41 | 1+(−4.5−7.79i)T+(−20.5+35.5i)T2 |
| 43 | 1+(−2.20−1.27i)T+(21.5+37.2i)T2 |
| 47 | 1+(9.43+5.44i)T+(23.5+40.7i)T2 |
| 53 | 1+3.55iT−53T2 |
| 59 | 1+(−2.72−4.71i)T+(−29.5+51.0i)T2 |
| 61 | 1+(4−6.92i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−0.301+0.174i)T+(33.5−58.0i)T2 |
| 71 | 1+13.3T+71T2 |
| 73 | 1−iT−73T2 |
| 79 | 1+(−8.34+14.4i)T+(−39.5−68.4i)T2 |
| 83 | 1+(−4.71−2.72i)T+(41.5+71.8i)T2 |
| 89 | 1+9T+89T2 |
| 97 | 1+(7.61+4.39i)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
−9.373442301791414041863858157950, −8.453505232763313032449521991846, −7.67200150899748574084827668781, −7.03425187799792393197020344952, −5.99030135531057856854626751787, −4.85561133051351852743234338798, −4.05124396753137062726543635823, −2.64111953416455343705167776431, −1.91121521712688236787892436783, −0.10246353154306206935961105424,
1.51733003721263135997844622854, 2.80411736353517922731003428963, 3.99685387688611163999115370024, 5.11141767083487073126422444153, 6.12498154022965176259003090909, 6.56416748692617554562181564673, 7.86835728255680159585753695233, 8.339490578109850870881137555081, 8.917825423181372554618526657774, 10.04182254225589312621368954788