L(s) = 1 | + (−0.310 − 0.537i)5-s + (−1.44 + 2.21i)7-s + (−1.91 − 1.10i)11-s − 0.963i·13-s + (0.653 − 1.13i)17-s + (−1.57 + 0.911i)19-s + (5.27 − 3.04i)23-s + (2.30 − 3.99i)25-s − 6.68i·29-s + (3.48 + 2.01i)31-s + (1.63 + 0.0859i)35-s + (0.165 + 0.286i)37-s + 11.7·41-s − 2.18·43-s + (−1.27 − 2.20i)47-s + ⋯ |
L(s) = 1 | + (−0.138 − 0.240i)5-s + (−0.544 + 0.838i)7-s + (−0.577 − 0.333i)11-s − 0.267i·13-s + (0.158 − 0.274i)17-s + (−0.362 + 0.209i)19-s + (1.10 − 0.635i)23-s + (0.461 − 0.799i)25-s − 1.24i·29-s + (0.626 + 0.361i)31-s + (0.277 + 0.0145i)35-s + (0.0271 + 0.0470i)37-s + 1.83·41-s − 0.333·43-s + (−0.185 − 0.321i)47-s + ⋯ |
Λ(s)=(=(1512s/2ΓC(s)L(s)(0.562+0.826i)Λ(2−s)
Λ(s)=(=(1512s/2ΓC(s+1/2)L(s)(0.562+0.826i)Λ(1−s)
Degree: |
2 |
Conductor: |
1512
= 23⋅33⋅7
|
Sign: |
0.562+0.826i
|
Analytic conductor: |
12.0733 |
Root analytic conductor: |
3.47467 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1512(1025,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1512, ( :1/2), 0.562+0.826i)
|
Particular Values
L(1) |
≈ |
1.280965694 |
L(21) |
≈ |
1.280965694 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1+(1.44−2.21i)T |
good | 5 | 1+(0.310+0.537i)T+(−2.5+4.33i)T2 |
| 11 | 1+(1.91+1.10i)T+(5.5+9.52i)T2 |
| 13 | 1+0.963iT−13T2 |
| 17 | 1+(−0.653+1.13i)T+(−8.5−14.7i)T2 |
| 19 | 1+(1.57−0.911i)T+(9.5−16.4i)T2 |
| 23 | 1+(−5.27+3.04i)T+(11.5−19.9i)T2 |
| 29 | 1+6.68iT−29T2 |
| 31 | 1+(−3.48−2.01i)T+(15.5+26.8i)T2 |
| 37 | 1+(−0.165−0.286i)T+(−18.5+32.0i)T2 |
| 41 | 1−11.7T+41T2 |
| 43 | 1+2.18T+43T2 |
| 47 | 1+(1.27+2.20i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−0.985−0.569i)T+(26.5+45.8i)T2 |
| 59 | 1+(−6.20+10.7i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−0.603+0.348i)T+(30.5−52.8i)T2 |
| 67 | 1+(3.61−6.25i)T+(−33.5−58.0i)T2 |
| 71 | 1+7.40iT−71T2 |
| 73 | 1+(2.17+1.25i)T+(36.5+63.2i)T2 |
| 79 | 1+(−4.52−7.83i)T+(−39.5+68.4i)T2 |
| 83 | 1−8.85T+83T2 |
| 89 | 1+(4.20+7.29i)T+(−44.5+77.0i)T2 |
| 97 | 1+12.4iT−97T2 |
show more | |
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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
−9.308191977157950065148828971117, −8.512390346242674387545149647083, −7.956868164028889887016063385595, −6.81721673417968216360257163884, −6.06505872703600359341276182556, −5.25125429954569123120304246373, −4.35360590919092273018400808219, −3.09150311924977690435047122635, −2.37546869996277260073052503565, −0.58512265967090040623169241082,
1.13568785292751868095358800419, 2.66823649793337296276859810215, 3.56539094117109623514778721157, 4.51653513764435873193846852627, 5.44286516128832762601182208795, 6.53065945742351604049675765866, 7.19914196339591648034661148173, 7.80077840701541208751959642316, 8.939731305351096913318369641953, 9.560050896761407919446574184718