L(s) = 1 | + (−0.260 − 1.14i)3-s + (−1.85 − 2.32i)5-s + (0.115 + 0.507i)7-s + (1.46 − 0.707i)9-s + (−0.585 − 0.281i)11-s + (−0.444 − 0.214i)13-s + (−2.16 + 2.72i)15-s − 7.42·17-s + (1.47 − 6.46i)19-s + (0.549 − 0.264i)21-s + (−4.74 + 5.95i)23-s + (−0.853 + 3.73i)25-s + (−3.37 − 4.23i)27-s + (−4.56 + 2.85i)29-s + (−3.72 − 4.67i)31-s + ⋯ |
L(s) = 1 | + (−0.150 − 0.658i)3-s + (−0.828 − 1.03i)5-s + (0.0438 + 0.191i)7-s + (0.489 − 0.235i)9-s + (−0.176 − 0.0849i)11-s + (−0.123 − 0.0593i)13-s + (−0.560 + 0.702i)15-s − 1.79·17-s + (0.338 − 1.48i)19-s + (0.119 − 0.0577i)21-s + (−0.990 + 1.24i)23-s + (−0.170 + 0.747i)25-s + (−0.650 − 0.815i)27-s + (−0.848 + 0.529i)29-s + (−0.669 − 0.839i)31-s + ⋯ |
Λ(s)=(=(464s/2ΓC(s)L(s)(−0.912+0.408i)Λ(2−s)
Λ(s)=(=(464s/2ΓC(s+1/2)L(s)(−0.912+0.408i)Λ(1−s)
Degree: |
2 |
Conductor: |
464
= 24⋅29
|
Sign: |
−0.912+0.408i
|
Analytic conductor: |
3.70505 |
Root analytic conductor: |
1.92485 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ464(401,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 464, ( :1/2), −0.912+0.408i)
|
Particular Values
L(1) |
≈ |
0.162410−0.760759i |
L(21) |
≈ |
0.162410−0.760759i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 29 | 1+(4.56−2.85i)T |
good | 3 | 1+(0.260+1.14i)T+(−2.70+1.30i)T2 |
| 5 | 1+(1.85+2.32i)T+(−1.11+4.87i)T2 |
| 7 | 1+(−0.115−0.507i)T+(−6.30+3.03i)T2 |
| 11 | 1+(0.585+0.281i)T+(6.85+8.60i)T2 |
| 13 | 1+(0.444+0.214i)T+(8.10+10.1i)T2 |
| 17 | 1+7.42T+17T2 |
| 19 | 1+(−1.47+6.46i)T+(−17.1−8.24i)T2 |
| 23 | 1+(4.74−5.95i)T+(−5.11−22.4i)T2 |
| 31 | 1+(3.72+4.67i)T+(−6.89+30.2i)T2 |
| 37 | 1+(−2.23+1.07i)T+(23.0−28.9i)T2 |
| 41 | 1−7.82T+41T2 |
| 43 | 1+(0.404−0.507i)T+(−9.56−41.9i)T2 |
| 47 | 1+(−7.92−3.81i)T+(29.3+36.7i)T2 |
| 53 | 1+(0.717+0.900i)T+(−11.7+51.6i)T2 |
| 59 | 1−5.31T+59T2 |
| 61 | 1+(2.15+9.45i)T+(−54.9+26.4i)T2 |
| 67 | 1+(−4.07+1.96i)T+(41.7−52.3i)T2 |
| 71 | 1+(12.7+6.13i)T+(44.2+55.5i)T2 |
| 73 | 1+(−5.44+6.83i)T+(−16.2−71.1i)T2 |
| 79 | 1+(−3.71+1.79i)T+(49.2−61.7i)T2 |
| 83 | 1+(−0.952+4.17i)T+(−74.7−36.0i)T2 |
| 89 | 1+(0.743+0.932i)T+(−19.8+86.7i)T2 |
| 97 | 1+(2.61−11.4i)T+(−87.3−42.0i)T2 |
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
−10.99776300677444350056689984198, −9.430752185586944993348464289958, −8.922915627958788897714613812710, −7.76254363311651659212716298374, −7.18020455962660104871593326384, −5.97634312060132025465166505691, −4.77418925389824489925646981059, −3.92719457347048808590704445860, −2.09521803030342607967663268076, −0.47611251103397378609378223756,
2.32200214017695639549209058825, 3.85069132983340065706492128039, 4.35028400240682989491347827162, 5.82219541137133643050771649057, 6.98497727269706942742092393336, 7.63516553577144611125311647265, 8.739227348709638614588196022953, 9.934228432399595703869359830073, 10.61582280136298210064350630735, 11.14645252891245087625577285875