L(s) = 1 | + (0.294 − 0.955i)3-s + (0.365 − 0.930i)5-s + (−0.826 − 0.563i)9-s + (0.563 + 0.826i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)15-s + (−0.866 − 0.5i)17-s + (−0.294 − 0.955i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (−0.781 + 0.623i)27-s + (0.149 + 0.988i)31-s + (0.955 − 0.294i)33-s + (−0.563 + 0.826i)37-s + (−0.680 + 0.733i)39-s + ⋯ |
L(s) = 1 | + (0.294 − 0.955i)3-s + (0.365 − 0.930i)5-s + (−0.826 − 0.563i)9-s + (0.563 + 0.826i)11-s + (−0.900 − 0.433i)13-s + (−0.781 − 0.623i)15-s + (−0.866 − 0.5i)17-s + (−0.294 − 0.955i)19-s + (0.988 + 0.149i)23-s + (−0.733 − 0.680i)25-s + (−0.781 + 0.623i)27-s + (0.149 + 0.988i)31-s + (0.955 − 0.294i)33-s + (−0.563 + 0.826i)37-s + (−0.680 + 0.733i)39-s + ⋯ |
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.214+0.976i)Λ(1−s)
Λ(s)=(=(812s/2ΓR(s+1)L(s)(−0.214+0.976i)Λ(1−s)
Degree: |
1 |
Conductor: |
812
= 22⋅7⋅29
|
Sign: |
−0.214+0.976i
|
Analytic conductor: |
87.2615 |
Root analytic conductor: |
87.2615 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ812(243,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 812, (1: ), −0.214+0.976i)
|
Particular Values
L(21) |
≈ |
−0.2013500203−0.2502489205i |
L(21) |
≈ |
−0.2013500203−0.2502489205i |
L(1) |
≈ |
0.8434885419−0.4933431742i |
L(1) |
≈ |
0.8434885419−0.4933431742i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1 |
| 29 | 1 |
good | 3 | 1+(0.294−0.955i)T |
| 5 | 1+(0.365−0.930i)T |
| 11 | 1+(0.563+0.826i)T |
| 13 | 1+(−0.900−0.433i)T |
| 17 | 1+(−0.866−0.5i)T |
| 19 | 1+(−0.294−0.955i)T |
| 23 | 1+(0.988+0.149i)T |
| 31 | 1+(0.149+0.988i)T |
| 37 | 1+(−0.563+0.826i)T |
| 41 | 1−iT |
| 43 | 1+(0.781+0.623i)T |
| 47 | 1+(−0.997+0.0747i)T |
| 53 | 1+(−0.988+0.149i)T |
| 59 | 1+(−0.5+0.866i)T |
| 61 | 1+(−0.680−0.733i)T |
| 67 | 1+(0.0747−0.997i)T |
| 71 | 1+(−0.900−0.433i)T |
| 73 | 1+(0.930−0.365i)T |
| 79 | 1+(−0.563+0.826i)T |
| 83 | 1+(−0.222+0.974i)T |
| 89 | 1+(0.930+0.365i)T |
| 97 | 1+(0.974+0.222i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.42150261289116433951716986287, −21.729835182140519134977456824603, −21.26897847976813111594894412165, −20.264462903691636261201523374047, −19.23949929522886831582388098777, −18.93858115005090590133640370998, −17.5544732759578352101041496086, −16.98159787636682462685423503337, −16.15132945560390418196962283623, −15.12013327136661836456762476771, −14.57490205608871363592308422044, −13.98793299078833623271614216060, −12.977966235983025325789269254806, −11.628280467793767420680636593084, −11.00539496018060137758414687282, −10.238990738999469995274517290244, −9.431687799385288336914770190450, −8.68068365772875435907132825438, −7.60340466819269968345812464390, −6.49529656782462054229292715869, −5.75960176092118596984524849992, −4.58155115788246871320582449542, −3.699977793270190239068792096159, −2.83168589902527916335503368815, −1.874269123939786087565544868003,
0.0654011970912102267379290472, 1.16745060385258429266162570767, 2.087048844481937085056758479845, 3.014832928964070791887704789928, 4.56288639310792955926530500185, 5.17712790279458094321215529846, 6.48267463915958129421807572792, 7.10071308817008724886481550173, 8.068728270474876747056677704223, 9.07693680592202444959066548266, 9.447960229322068067906931248234, 10.83040604131497864131624877806, 11.95785021237972086377497893386, 12.50746285729018526835201687308, 13.217041213984985325235593247943, 13.952322102164775488361876619718, 14.90607650649822718938203530521, 15.70166686396715700066119950847, 17.02474187942670876683266145942, 17.43206920721050289638344226107, 18.04421560401213393481837495145, 19.3062024795148015275899100818, 19.82489611418460774844597646402, 20.423393458801744613169937667713, 21.3294345606345833524948951874