L(s) = 1 | + (−1.30 − 2.26i)2-s + (−2.42 + 4.20i)4-s + (1.11 + 1.93i)5-s + (−2 + 1.73i)7-s + 7.47·8-s + (2.92 − 5.06i)10-s + (−1.5 + 2.59i)11-s − 13-s + (6.54 + 2.26i)14-s + (−4.92 − 8.53i)16-s + (0.736 − 1.27i)17-s + (−1.5 − 2.59i)19-s − 10.8·20-s + 7.85·22-s + (−4.11 − 7.13i)23-s + ⋯ |
L(s) = 1 | + (−0.925 − 1.60i)2-s + (−1.21 + 2.10i)4-s + (0.499 + 0.866i)5-s + (−0.755 + 0.654i)7-s + 2.64·8-s + (0.925 − 1.60i)10-s + (−0.452 + 0.783i)11-s − 0.277·13-s + (1.74 + 0.605i)14-s + (−1.23 − 2.13i)16-s + (0.178 − 0.309i)17-s + (−0.344 − 0.596i)19-s − 2.42·20-s + 1.67·22-s + (−0.858 − 1.48i)23-s + ⋯ |
Λ(s)=(=(819s/2ΓC(s)L(s)(−0.605−0.795i)Λ(2−s)
Λ(s)=(=(819s/2ΓC(s+1/2)L(s)(−0.605−0.795i)Λ(1−s)
Degree: |
2 |
Conductor: |
819
= 32⋅7⋅13
|
Sign: |
−0.605−0.795i
|
Analytic conductor: |
6.53974 |
Root analytic conductor: |
2.55729 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ819(235,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
1
|
Selberg data: |
(2, 819, ( :1/2), −0.605−0.795i)
|
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(2−1.73i)T |
| 13 | 1+T |
good | 2 | 1+(1.30+2.26i)T+(−1+1.73i)T2 |
| 5 | 1+(−1.11−1.93i)T+(−2.5+4.33i)T2 |
| 11 | 1+(1.5−2.59i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−0.736+1.27i)T+(−8.5−14.7i)T2 |
| 19 | 1+(1.5+2.59i)T+(−9.5+16.4i)T2 |
| 23 | 1+(4.11+7.13i)T+(−11.5+19.9i)T2 |
| 29 | 1+4.47T+29T2 |
| 31 | 1+(2.5−4.33i)T+(−15.5−26.8i)T2 |
| 37 | 1+(2.35+4.07i)T+(−18.5+32.0i)T2 |
| 41 | 1−4.47T+41T2 |
| 43 | 1+8T+43T2 |
| 47 | 1+(3.73+6.47i)T+(−23.5+40.7i)T2 |
| 53 | 1+(3.73−6.47i)T+(−26.5−45.8i)T2 |
| 59 | 1+(0.736−1.27i)T+(−29.5−51.0i)T2 |
| 61 | 1+(1.5+2.59i)T+(−30.5+52.8i)T2 |
| 67 | 1+(−1.5+2.59i)T+(−33.5−58.0i)T2 |
| 71 | 1−8.94T+71T2 |
| 73 | 1+(−1.35+2.34i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−1.35−2.34i)T+(−39.5+68.4i)T2 |
| 83 | 1+83T2 |
| 89 | 1+(−1.11−1.93i)T+(−44.5+77.0i)T2 |
| 97 | 1−9.41T+97T2 |
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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
−9.876489472587370965440491531903, −9.213795769930802814763199912084, −8.379930000021349043469553804432, −7.29600919162277366229111860774, −6.38106465960017753553064942877, −4.92359572477605806208166223835, −3.61803831127645157847767493753, −2.60344181598290445483504774027, −2.10763991469750759528048455278, 0,
1.44753318114864922335316415897, 3.72307114132072439895375022139, 5.07413648051854586608028194205, 5.78329784947620868191492817450, 6.44397900283482383999943819474, 7.55980208973122466636735299024, 8.071750805877635202654697591420, 9.022596016919218622266460582322, 9.685189091288742995533559344134