L(s) = 1 | + (−1.12 − 0.862i)2-s + (0.418 + 1.68i)3-s + (0.511 + 1.93i)4-s + (−1.60 + 2.78i)5-s + (0.980 − 2.24i)6-s + (1.82 − 1.05i)7-s + (1.09 − 2.60i)8-s + (−2.64 + 1.40i)9-s + (4.20 − 1.73i)10-s + (3.47 − 2.00i)11-s + (−3.03 + 1.66i)12-s + (−0.341 − 0.197i)13-s + (−2.94 − 0.392i)14-s + (−5.35 − 1.53i)15-s + (−3.47 + 1.97i)16-s − 1.20i·17-s + ⋯ |
L(s) = 1 | + (−0.792 − 0.609i)2-s + (0.241 + 0.970i)3-s + (0.255 + 0.966i)4-s + (−0.719 + 1.24i)5-s + (0.400 − 0.916i)6-s + (0.688 − 0.397i)7-s + (0.386 − 0.922i)8-s + (−0.883 + 0.469i)9-s + (1.33 − 0.548i)10-s + (1.04 − 0.605i)11-s + (−0.876 + 0.481i)12-s + (−0.0948 − 0.0547i)13-s + (−0.788 − 0.105i)14-s + (−1.38 − 0.397i)15-s + (−0.869 + 0.494i)16-s − 0.292i·17-s + ⋯ |
Λ(s)=(=(72s/2ΓC(s)L(s)(0.704−0.709i)Λ(2−s)
Λ(s)=(=(72s/2ΓC(s+1/2)L(s)(0.704−0.709i)Λ(1−s)
Degree: |
2 |
Conductor: |
72
= 23⋅32
|
Sign: |
0.704−0.709i
|
Analytic conductor: |
0.574922 |
Root analytic conductor: |
0.758236 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ72(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 72, ( :1/2), 0.704−0.709i)
|
Particular Values
L(1) |
≈ |
0.622182+0.259016i |
L(21) |
≈ |
0.622182+0.259016i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.12+0.862i)T |
| 3 | 1+(−0.418−1.68i)T |
good | 5 | 1+(1.60−2.78i)T+(−2.5−4.33i)T2 |
| 7 | 1+(−1.82+1.05i)T+(3.5−6.06i)T2 |
| 11 | 1+(−3.47+2.00i)T+(5.5−9.52i)T2 |
| 13 | 1+(0.341+0.197i)T+(6.5+11.2i)T2 |
| 17 | 1+1.20iT−17T2 |
| 19 | 1+1.62T+19T2 |
| 23 | 1+(−2.74+4.75i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−2.95−5.12i)T+(−14.5+25.1i)T2 |
| 31 | 1+(−3.34−1.93i)T+(15.5+26.8i)T2 |
| 37 | 1+10.8iT−37T2 |
| 41 | 1+(1.23+0.715i)T+(20.5+35.5i)T2 |
| 43 | 1+(1.21+2.10i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−0.792−1.37i)T+(−23.5+40.7i)T2 |
| 53 | 1+7.07T+53T2 |
| 59 | 1+(2.29+1.32i)T+(29.5+51.0i)T2 |
| 61 | 1+(8.18−4.72i)T+(30.5−52.8i)T2 |
| 67 | 1+(2.60−4.51i)T+(−33.5−58.0i)T2 |
| 71 | 1+2.69T+71T2 |
| 73 | 1−9.49T+73T2 |
| 79 | 1+(−1.53+0.886i)T+(39.5−68.4i)T2 |
| 83 | 1+(1.30−0.755i)T+(41.5−71.8i)T2 |
| 89 | 1−11.2iT−89T2 |
| 97 | 1+(−5.84−10.1i)T+(−48.5+84.0i)T2 |
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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
−14.76995308394917264781304323831, −14.05476594723919265809422397691, −12.05301158337697153316303626418, −10.92554107906035083042504973035, −10.71377736334231792761570862817, −9.190396908295398497939537176916, −8.119380102612132859704638094277, −6.82991345986324474356725257639, −4.22135047638566990136498126202, −3.03718913967386118541571283882,
1.45928096222127515576478818258, 4.80328813773384434714049220641, 6.41350412703383057733337968514, 7.79855447360569161300075210335, 8.486589654666224307559450998802, 9.451555866011288059287021841224, 11.50974830539407792799566364488, 12.12769226139601447743213715056, 13.50027935289146691638728829602, 14.76443171591892906531114388408