L(s) = 1 | + (0.427 − 0.903i)3-s + (0.989 + 0.146i)5-s + (−0.0980 − 0.995i)7-s + (−0.634 − 0.773i)9-s + (0.998 − 0.0490i)11-s + (−0.803 − 0.595i)13-s + (0.555 − 0.831i)15-s + (−0.555 − 0.831i)17-s + (−0.970 − 0.242i)19-s + (−0.941 − 0.336i)21-s + (−0.471 + 0.881i)23-s + (0.956 + 0.290i)25-s + (−0.970 + 0.242i)27-s + (0.740 − 0.671i)29-s + (0.382 + 0.923i)31-s + ⋯ |
L(s) = 1 | + (0.427 − 0.903i)3-s + (0.989 + 0.146i)5-s + (−0.0980 − 0.995i)7-s + (−0.634 − 0.773i)9-s + (0.998 − 0.0490i)11-s + (−0.803 − 0.595i)13-s + (0.555 − 0.831i)15-s + (−0.555 − 0.831i)17-s + (−0.970 − 0.242i)19-s + (−0.941 − 0.336i)21-s + (−0.471 + 0.881i)23-s + (0.956 + 0.290i)25-s + (−0.970 + 0.242i)27-s + (0.740 − 0.671i)29-s + (0.382 + 0.923i)31-s + ⋯ |
Λ(s)=(=(512s/2ΓR(s)L(s)(−0.302−0.953i)Λ(1−s)
Λ(s)=(=(512s/2ΓR(s)L(s)(−0.302−0.953i)Λ(1−s)
Degree: |
1 |
Conductor: |
512
= 29
|
Sign: |
−0.302−0.953i
|
Analytic conductor: |
2.37771 |
Root analytic conductor: |
2.37771 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ512(125,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 512, (0: ), −0.302−0.953i)
|
Particular Values
L(21) |
≈ |
1.004971670−1.372569813i |
L(21) |
≈ |
1.004971670−1.372569813i |
L(1) |
≈ |
1.174435627−0.6407913774i |
L(1) |
≈ |
1.174435627−0.6407913774i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
good | 3 | 1+(0.427−0.903i)T |
| 5 | 1+(0.989+0.146i)T |
| 7 | 1+(−0.0980−0.995i)T |
| 11 | 1+(0.998−0.0490i)T |
| 13 | 1+(−0.803−0.595i)T |
| 17 | 1+(−0.555−0.831i)T |
| 19 | 1+(−0.970−0.242i)T |
| 23 | 1+(−0.471+0.881i)T |
| 29 | 1+(0.740−0.671i)T |
| 31 | 1+(0.382+0.923i)T |
| 37 | 1+(−0.857−0.514i)T |
| 41 | 1+(0.956−0.290i)T |
| 43 | 1+(−0.903+0.427i)T |
| 47 | 1+(0.980+0.195i)T |
| 53 | 1+(0.740+0.671i)T |
| 59 | 1+(0.803−0.595i)T |
| 61 | 1+(0.941−0.336i)T |
| 67 | 1+(−0.336−0.941i)T |
| 71 | 1+(0.773+0.634i)T |
| 73 | 1+(−0.0980+0.995i)T |
| 79 | 1+(0.195+0.980i)T |
| 83 | 1+(−0.857+0.514i)T |
| 89 | 1+(−0.471−0.881i)T |
| 97 | 1+(−0.923+0.382i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.107540613496022834928757610552, −22.49764382959435232683655324042, −22.00239265273452632433976585708, −21.4724827889337374198251207475, −20.65945655064078326097749521951, −19.62159070131330197409183674028, −18.9721444676795774852916997194, −17.71481001927232259889314122356, −16.95035256467872780192576894848, −16.27516863192200375295769680632, −14.994695910933014606453216174839, −14.65898482886790870637038035814, −13.70989203395122211169645123127, −12.60278159048517137425179290644, −11.72850082599226708126557707497, −10.482833384733538271622140181945, −9.80499188504763249182269879163, −8.87270680616835784431337275780, −8.50478843649648898863403153896, −6.68377672082373113725711487017, −5.91657682619228304523038206790, −4.84164221603122405418926213504, −3.98767405409493199425530866759, −2.54109731253532145269442654813, −1.9310108169926703448197947861,
0.85512621008299191386762486486, 1.989827704646080875748923653220, 2.94747948997248697506228377993, 4.199316188680660672854245671325, 5.5506895267882805545229167512, 6.69641820311182170364001741373, 7.07124446228437158856521442441, 8.29692666024639884533470932804, 9.31115222679207138246756592892, 10.07758379684350422030597810347, 11.18521549869772103692028985591, 12.28719856041952176274116113246, 13.11962036815663320879185831005, 13.991809852604931210704435224575, 14.277818509465705533935258904228, 15.51611725780296323315318785328, 16.98526276487896643677542729162, 17.42786560952100421589582343450, 18.05692590752419400743388321947, 19.38126012005734775803032632166, 19.756418604610038513161087829995, 20.6685696735116203106538544877, 21.6454742395536681132033637194, 22.61772489825173202219255643249, 23.3272725069369910272538102596