L(s) = 1 | + (0.874 − 1.11i)2-s + (−0.707 + 0.707i)3-s + (−0.470 − 1.94i)4-s + (−0.334 − 0.334i)5-s + (0.167 + 1.40i)6-s + 4.55i·7-s + (−2.57 − 1.17i)8-s − 1.00i·9-s + (−0.665 + 0.0793i)10-s + (−2.47 − 2.47i)11-s + (1.70 + 1.04i)12-s + (−0.0594 + 0.0594i)13-s + (5.06 + 3.98i)14-s + 0.473·15-s + (−3.55 + 1.82i)16-s + 3.61·17-s + ⋯ |
L(s) = 1 | + (0.618 − 0.785i)2-s + (−0.408 + 0.408i)3-s + (−0.235 − 0.971i)4-s + (−0.149 − 0.149i)5-s + (0.0683 + 0.573i)6-s + 1.72i·7-s + (−0.909 − 0.416i)8-s − 0.333i·9-s + (−0.210 + 0.0250i)10-s + (−0.745 − 0.745i)11-s + (0.492 + 0.300i)12-s + (−0.0164 + 0.0164i)13-s + (1.35 + 1.06i)14-s + 0.122·15-s + (−0.889 + 0.457i)16-s + 0.877·17-s + ⋯ |
Λ(s)=(=(48s/2ΓC(s)L(s)(0.762+0.646i)Λ(2−s)
Λ(s)=(=(48s/2ΓC(s+1/2)L(s)(0.762+0.646i)Λ(1−s)
Degree: |
2 |
Conductor: |
48
= 24⋅3
|
Sign: |
0.762+0.646i
|
Analytic conductor: |
0.383281 |
Root analytic conductor: |
0.619097 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ48(37,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 48, ( :1/2), 0.762+0.646i)
|
Particular Values
L(1) |
≈ |
0.859126−0.315241i |
L(21) |
≈ |
0.859126−0.315241i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.874+1.11i)T |
| 3 | 1+(0.707−0.707i)T |
good | 5 | 1+(0.334+0.334i)T+5iT2 |
| 7 | 1−4.55iT−7T2 |
| 11 | 1+(2.47+2.47i)T+11iT2 |
| 13 | 1+(0.0594−0.0594i)T−13iT2 |
| 17 | 1−3.61T+17T2 |
| 19 | 1+(−2.55+2.55i)T−19iT2 |
| 23 | 1+2.82iT−23T2 |
| 29 | 1+(5.16−5.16i)T−29iT2 |
| 31 | 1+0.557T+31T2 |
| 37 | 1+(−4.38−4.38i)T+37iT2 |
| 41 | 1−9.27iT−41T2 |
| 43 | 1+(1.61+1.61i)T+43iT2 |
| 47 | 1−2.82T+47T2 |
| 53 | 1+(0.493+0.493i)T+53iT2 |
| 59 | 1+(−4−4i)T+59iT2 |
| 61 | 1+(−2.72+2.72i)T−61iT2 |
| 67 | 1+(−3.77+3.77i)T−67iT2 |
| 71 | 1+9.11iT−71T2 |
| 73 | 1−0.541iT−73T2 |
| 79 | 1+10.9T+79T2 |
| 83 | 1+(10.6−10.6i)T−83iT2 |
| 89 | 1+14.6iT−89T2 |
| 97 | 1−4.31T+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
−15.45259046146594049541990349980, −14.48970710928985023142069963813, −12.97197468381363123603374713277, −12.03702935072545998173445842602, −11.16531769844445068984838941454, −9.800984711243321477277656143961, −8.596318950109452418931733201385, −5.97148374437644136007961728477, −5.04018784539916012716801928425, −2.93277313035644301932611142198,
3.88431800021340996070071180644, 5.50016234533184009125213007544, 7.25469706611507216003136633047, 7.66130997167112872618999970533, 9.954645857861525156894588031065, 11.36083120799465652053750283849, 12.73589645070448832286116986788, 13.57402103460834438797867056377, 14.57766008330093553152231520766, 15.89760953308572664773995942205