L(s) = 1 | + (0.720 + 1.24i)2-s + (2.96 − 5.12i)4-s + (3.30 + 5.71i)5-s + (−10.3 − 15.3i)7-s + 20.0·8-s + (−4.76 + 8.24i)10-s + (14.5 − 25.2i)11-s + 16.7·13-s + (11.7 − 23.9i)14-s + (−9.21 − 15.9i)16-s + (23.5 − 40.7i)17-s + (−1.71 − 2.97i)19-s + 39.0·20-s + 42.0·22-s + (−14.7 − 25.5i)23-s + ⋯ |
L(s) = 1 | + (0.254 + 0.441i)2-s + (0.370 − 0.640i)4-s + (0.295 + 0.511i)5-s + (−0.557 − 0.830i)7-s + 0.887·8-s + (−0.150 + 0.260i)10-s + (0.399 − 0.692i)11-s + 0.358·13-s + (0.224 − 0.457i)14-s + (−0.143 − 0.249i)16-s + (0.336 − 0.582i)17-s + (−0.0207 − 0.0358i)19-s + 0.437·20-s + 0.407·22-s + (−0.133 − 0.231i)23-s + ⋯ |
Λ(s)=(=(189s/2ΓC(s)L(s)(0.868+0.496i)Λ(4−s)
Λ(s)=(=(189s/2ΓC(s+3/2)L(s)(0.868+0.496i)Λ(1−s)
Degree: |
2 |
Conductor: |
189
= 33⋅7
|
Sign: |
0.868+0.496i
|
Analytic conductor: |
11.1513 |
Root analytic conductor: |
3.33936 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ189(109,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 189, ( :3/2), 0.868+0.496i)
|
Particular Values
L(2) |
≈ |
2.17834−0.579007i |
L(21) |
≈ |
2.17834−0.579007i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(10.3+15.3i)T |
good | 2 | 1+(−0.720−1.24i)T+(−4+6.92i)T2 |
| 5 | 1+(−3.30−5.71i)T+(−62.5+108.i)T2 |
| 11 | 1+(−14.5+25.2i)T+(−665.5−1.15e3i)T2 |
| 13 | 1−16.7T+2.19e3T2 |
| 17 | 1+(−23.5+40.7i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(1.71+2.97i)T+(−3.42e3+5.94e3i)T2 |
| 23 | 1+(14.7+25.5i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1−223.T+2.43e4T2 |
| 31 | 1+(60.3−104.i)T+(−1.48e4−2.57e4i)T2 |
| 37 | 1+(17.4+30.2i)T+(−2.53e4+4.38e4i)T2 |
| 41 | 1−192.T+6.89e4T2 |
| 43 | 1+5.61T+7.95e4T2 |
| 47 | 1+(−179.−311.i)T+(−5.19e4+8.99e4i)T2 |
| 53 | 1+(16.6−28.8i)T+(−7.44e4−1.28e5i)T2 |
| 59 | 1+(371.−642.i)T+(−1.02e5−1.77e5i)T2 |
| 61 | 1+(329.+570.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(470.−815.i)T+(−1.50e5−2.60e5i)T2 |
| 71 | 1+871.T+3.57e5T2 |
| 73 | 1+(−366.+634.i)T+(−1.94e5−3.36e5i)T2 |
| 79 | 1+(671.+1.16e3i)T+(−2.46e5+4.26e5i)T2 |
| 83 | 1+588.T+5.71e5T2 |
| 89 | 1+(−623.−1.08e3i)T+(−3.52e5+6.10e5i)T2 |
| 97 | 1−691.T+9.12e5T2 |
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
−11.97800103358481980175745686881, −10.73885002900419864883121646606, −10.33833784705689080251722226550, −9.119957595013647755909740152561, −7.60114373634617714771181527346, −6.61992959108208189934151661955, −5.97197593200055355605303962505, −4.49088899808930372812078476146, −2.94882873696551600990685333404, −1.00726866050985513011172058623,
1.71864709085955492239094646098, 3.07803439195255393556716710340, 4.38487754429386478855983560750, 5.80486365166671159107925388233, 6.97940494684330358792375670861, 8.250333342638481778838835748175, 9.203974562690957666985233391497, 10.26861481255553736721064619953, 11.47429001795464660900243950591, 12.35612231912170113953198643694