L(s) = 1 | + (−0.132 + 0.229i)2-s + (1.45 + 2.51i)3-s + (0.964 + 1.67i)4-s + (−0.717 + 1.24i)5-s − 0.769·6-s − 1.03·8-s + (−2.73 + 4.73i)9-s + (−0.189 − 0.328i)10-s + (−2.75 − 4.76i)11-s + (−2.80 + 4.86i)12-s − 13-s − 4.17·15-s + (−1.79 + 3.10i)16-s + (2.41 + 4.18i)17-s + (−0.722 − 1.25i)18-s + (1.41 − 2.44i)19-s + ⋯ |
L(s) = 1 | + (−0.0935 + 0.162i)2-s + (0.839 + 1.45i)3-s + (0.482 + 0.835i)4-s + (−0.320 + 0.555i)5-s − 0.314·6-s − 0.367·8-s + (−0.910 + 1.57i)9-s + (−0.0600 − 0.104i)10-s + (−0.829 − 1.43i)11-s + (−0.810 + 1.40i)12-s − 0.277·13-s − 1.07·15-s + (−0.448 + 0.776i)16-s + (0.585 + 1.01i)17-s + (−0.170 − 0.295i)18-s + (0.323 − 0.560i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.947−0.318i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.947−0.318i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.947−0.318i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), −0.947−0.318i)
|
Particular Values
L(1) |
≈ |
0.283680+1.73456i |
L(21) |
≈ |
0.283680+1.73456i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+T |
good | 2 | 1+(0.132−0.229i)T+(−1−1.73i)T2 |
| 3 | 1+(−1.45−2.51i)T+(−1.5+2.59i)T2 |
| 5 | 1+(0.717−1.24i)T+(−2.5−4.33i)T2 |
| 11 | 1+(2.75+4.76i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−2.41−4.18i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−1.41+2.44i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−2.99+5.19i)T+(−11.5−19.9i)T2 |
| 29 | 1−1.04T+29T2 |
| 31 | 1+(−4.60−7.97i)T+(−15.5+26.8i)T2 |
| 37 | 1+(0.306−0.530i)T+(−18.5−32.0i)T2 |
| 41 | 1−10.6T+41T2 |
| 43 | 1+8.43T+43T2 |
| 47 | 1+(−1.20+2.08i)T+(−23.5−40.7i)T2 |
| 53 | 1+(−0.914−1.58i)T+(−26.5+45.8i)T2 |
| 59 | 1+(0.435+0.754i)T+(−29.5+51.0i)T2 |
| 61 | 1+(1.66−2.88i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−3.31−5.73i)T+(−33.5+58.0i)T2 |
| 71 | 1+6.85T+71T2 |
| 73 | 1+(−1.57−2.72i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−8.78+15.2i)T+(−39.5−68.4i)T2 |
| 83 | 1+11.4T+83T2 |
| 89 | 1+(0.497−0.861i)T+(−44.5−77.0i)T2 |
| 97 | 1+13.5T+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
−10.73050500409332091393694577257, −10.31698036940793143886941259988, −8.983621929688269994492001062833, −8.457636547255583003169658358700, −7.77220133842127776106111422981, −6.63335379523641607901984979996, −5.35814761682224137950660173257, −4.18568873221230792616114106612, −3.07265137464017317970966950054, −2.92159823608996315984967300542,
0.901102334752005866011637296960, 2.05109008566370431909845023807, 2.89711899091391201426487148685, 4.72665787897811243499847144843, 5.76053618575762177922543881669, 6.96159158066853666784036955359, 7.49703917631162851758779050304, 8.191074979375242046349419177395, 9.472702728691315638680248582441, 9.872479488697604737173516560567