L(s) = 1 | − i·2-s + (1.57 − 0.711i)3-s − 4-s + (0.396 + 2.20i)5-s + (−0.711 − 1.57i)6-s + (−1.80 − 4.35i)7-s + i·8-s + (1.98 − 2.24i)9-s + (2.20 − 0.396i)10-s + (−0.688 + 1.66i)11-s + (−1.57 + 0.711i)12-s + (3.54 − 3.54i)13-s + (−4.35 + 1.80i)14-s + (2.19 + 3.19i)15-s + 16-s + (−2.12 − 3.53i)17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.911 − 0.410i)3-s − 0.5·4-s + (0.177 + 0.984i)5-s + (−0.290 − 0.644i)6-s + (−0.682 − 1.64i)7-s + 0.353i·8-s + (0.662 − 0.748i)9-s + (0.695 − 0.125i)10-s + (−0.207 + 0.501i)11-s + (−0.455 + 0.205i)12-s + (0.982 − 0.982i)13-s + (−1.16 + 0.482i)14-s + (0.565 + 0.824i)15-s + 0.250·16-s + (−0.514 − 0.857i)17-s + ⋯ |
Λ(s)=(=(510s/2ΓC(s)L(s)(−0.241+0.970i)Λ(2−s)
Λ(s)=(=(510s/2ΓC(s+1/2)L(s)(−0.241+0.970i)Λ(1−s)
Degree: |
2 |
Conductor: |
510
= 2⋅3⋅5⋅17
|
Sign: |
−0.241+0.970i
|
Analytic conductor: |
4.07237 |
Root analytic conductor: |
2.01801 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ510(53,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 510, ( :1/2), −0.241+0.970i)
|
Particular Values
L(1) |
≈ |
1.08698−1.39057i |
L(21) |
≈ |
1.08698−1.39057i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+iT |
| 3 | 1+(−1.57+0.711i)T |
| 5 | 1+(−0.396−2.20i)T |
| 17 | 1+(2.12+3.53i)T |
good | 7 | 1+(1.80+4.35i)T+(−4.94+4.94i)T2 |
| 11 | 1+(0.688−1.66i)T+(−7.77−7.77i)T2 |
| 13 | 1+(−3.54+3.54i)T−13iT2 |
| 19 | 1+(−2.50+2.50i)T−19iT2 |
| 23 | 1+(−1.64+0.680i)T+(16.2−16.2i)T2 |
| 29 | 1+(−2.16−5.23i)T+(−20.5+20.5i)T2 |
| 31 | 1+(0.226−0.0939i)T+(21.9−21.9i)T2 |
| 37 | 1+(0.215−0.519i)T+(−26.1−26.1i)T2 |
| 41 | 1+(−10.5−4.37i)T+(28.9+28.9i)T2 |
| 43 | 1−3.98T+43T2 |
| 47 | 1+(3.45−3.45i)T−47iT2 |
| 53 | 1+11.3T+53T2 |
| 59 | 1+(6.94−6.94i)T−59iT2 |
| 61 | 1+(4.54−10.9i)T+(−43.1−43.1i)T2 |
| 67 | 1+(−9.46+9.46i)T−67iT2 |
| 71 | 1+(−0.247−0.597i)T+(−50.2+50.2i)T2 |
| 73 | 1+(5.67−13.7i)T+(−51.6−51.6i)T2 |
| 79 | 1+(−2.72+6.56i)T+(−55.8−55.8i)T2 |
| 83 | 1+0.153T+83T2 |
| 89 | 1−2.45iT−89T2 |
| 97 | 1+(−6.26−2.59i)T+(68.5+68.5i)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
−10.62450840389760946623154554535, −9.871915020729221559316020763051, −9.129691420242865302028354867269, −7.75393779511023771456740121439, −7.21305601976423467973050550645, −6.30724028310377584074731808854, −4.47470582379097273535422154066, −3.36670184239216378154209159126, −2.80425281433315149567928349051, −1.04253845529128544995101695822,
1.97038504260826417618524714276, 3.44055223014656738251829001227, 4.54707193962560089991148508179, 5.69874165165355919015035846831, 6.32599763652999363198948613650, 7.950199674419066588848324412370, 8.587658525775291478924151811763, 9.182413514724700152541048636113, 9.677128429420309201245652234834, 11.11999095898507773396879122874