L(s) = 1 | − 4i·7-s + 2·11-s − 4i·13-s + i·17-s + 5·19-s − 5i·23-s − 8·29-s + 7·31-s − 6i·37-s + 6·41-s + 2i·43-s + 8i·47-s − 9·49-s − 9i·53-s − 4·59-s + ⋯ |
L(s) = 1 | − 1.51i·7-s + 0.603·11-s − 1.10i·13-s + 0.242i·17-s + 1.14·19-s − 1.04i·23-s − 1.48·29-s + 1.25·31-s − 0.986i·37-s + 0.937·41-s + 0.304i·43-s + 1.16i·47-s − 1.28·49-s − 1.23i·53-s − 0.520·59-s + ⋯ |
Λ(s)=(=(5400s/2ΓC(s)L(s)(−0.447+0.894i)Λ(2−s)
Λ(s)=(=(5400s/2ΓC(s+1/2)L(s)(−0.447+0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
5400
= 23⋅33⋅52
|
Sign: |
−0.447+0.894i
|
Analytic conductor: |
43.1192 |
Root analytic conductor: |
6.56652 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ5400(649,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 5400, ( :1/2), −0.447+0.894i)
|
Particular Values
L(1) |
≈ |
1.832029897 |
L(21) |
≈ |
1.832029897 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1 |
good | 7 | 1+4iT−7T2 |
| 11 | 1−2T+11T2 |
| 13 | 1+4iT−13T2 |
| 17 | 1−iT−17T2 |
| 19 | 1−5T+19T2 |
| 23 | 1+5iT−23T2 |
| 29 | 1+8T+29T2 |
| 31 | 1−7T+31T2 |
| 37 | 1+6iT−37T2 |
| 41 | 1−6T+41T2 |
| 43 | 1−2iT−43T2 |
| 47 | 1−8iT−47T2 |
| 53 | 1+9iT−53T2 |
| 59 | 1+4T+59T2 |
| 61 | 1−13T+61T2 |
| 67 | 1+10iT−67T2 |
| 71 | 1+6T+71T2 |
| 73 | 1−6iT−73T2 |
| 79 | 1+9T+79T2 |
| 83 | 1−17iT−83T2 |
| 89 | 1−6T+89T2 |
| 97 | 1+8iT−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
−7.79593909787110178279249813167, −7.34332201287685559925364283471, −6.59068482433198771955878496084, −5.82533640025509501440969514072, −4.97266406245091826938568769950, −4.11691085566333923827286839761, −3.57829159514621225164626331239, −2.63152661745988196379328072350, −1.28528210237523047739542514516, −0.52823090543578266186422143339,
1.31353487338724060151061674662, 2.19025728196716371850517195257, 3.05355969380010303752950436859, 3.93053310083182628517750639645, 4.86075099394627336774032390671, 5.59999648962559276088828868777, 6.13141467073513282250848454377, 6.99970705041457890283100079190, 7.64503171774460451266334485033, 8.570934080550980413168953813353