L(s) = 1 | + (−2.22 + 1.74i)2-s + (4.99 − 1.43i)3-s + (1.93 − 7.76i)4-s + (10.2 − 4.55i)5-s + (−8.63 + 11.8i)6-s − 3.90·7-s + (9.22 + 20.6i)8-s + (22.9 − 14.2i)9-s + (−14.8 + 27.9i)10-s − 59.1i·11-s + (−1.45 − 41.5i)12-s − 63.6·13-s + (8.70 − 6.80i)14-s + (44.5 − 37.3i)15-s + (−56.5 − 29.9i)16-s + 69.6·17-s + ⋯ |
L(s) = 1 | + (−0.787 + 0.615i)2-s + (0.961 − 0.275i)3-s + (0.241 − 0.970i)4-s + (0.913 − 0.407i)5-s + (−0.587 + 0.808i)6-s − 0.210·7-s + (0.407 + 0.913i)8-s + (0.848 − 0.529i)9-s + (−0.468 + 0.883i)10-s − 1.62i·11-s + (−0.0350 − 0.999i)12-s − 1.35·13-s + (0.166 − 0.129i)14-s + (0.765 − 0.642i)15-s + (−0.883 − 0.468i)16-s + 0.993·17-s + ⋯ |
Λ(s)=(=(120s/2ΓC(s)L(s)(0.899+0.437i)Λ(4−s)
Λ(s)=(=(120s/2ΓC(s+3/2)L(s)(0.899+0.437i)Λ(1−s)
Degree: |
2 |
Conductor: |
120
= 23⋅3⋅5
|
Sign: |
0.899+0.437i
|
Analytic conductor: |
7.08022 |
Root analytic conductor: |
2.66087 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ120(59,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 120, ( :3/2), 0.899+0.437i)
|
Particular Values
L(2) |
≈ |
1.63700−0.377150i |
L(21) |
≈ |
1.63700−0.377150i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(2.22−1.74i)T |
| 3 | 1+(−4.99+1.43i)T |
| 5 | 1+(−10.2+4.55i)T |
good | 7 | 1+3.90T+343T2 |
| 11 | 1+59.1iT−1.33e3T2 |
| 13 | 1+63.6T+2.19e3T2 |
| 17 | 1−69.6T+4.91e3T2 |
| 19 | 1−33.0T+6.85e3T2 |
| 23 | 1−90.6iT−1.21e4T2 |
| 29 | 1−172.T+2.43e4T2 |
| 31 | 1−61.7iT−2.97e4T2 |
| 37 | 1+10.7T+5.06e4T2 |
| 41 | 1−475.iT−6.89e4T2 |
| 43 | 1+59.3iT−7.95e4T2 |
| 47 | 1+500.iT−1.03e5T2 |
| 53 | 1−407.iT−1.48e5T2 |
| 59 | 1+17.5iT−2.05e5T2 |
| 61 | 1−245.iT−2.26e5T2 |
| 67 | 1+35.7iT−3.00e5T2 |
| 71 | 1+889.T+3.57e5T2 |
| 73 | 1−617.iT−3.89e5T2 |
| 79 | 1−108.iT−4.93e5T2 |
| 83 | 1−628.T+5.71e5T2 |
| 89 | 1−763.iT−7.04e5T2 |
| 97 | 1−866.iT−9.12e5T2 |
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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
−13.33681925814338100377030419994, −11.92919976279237434685675203728, −10.25096383244273982131327264885, −9.561626541520517746616881881407, −8.632919703550976925991612707331, −7.69891618505883990576151267931, −6.42120987938071030070196638044, −5.23016225755894433842262366308, −2.84992829403550152452827838066, −1.14209744345640055116672406334,
1.93594762501469443151782267647, 2.93539915300381312828840112577, 4.67435938180775282412564267015, 6.93630519622823263543105364278, 7.75192951867439789143875339914, 9.225869662039419968322632749915, 9.904106841577495052725996698224, 10.38133382949822126844191902628, 12.19268792440616040104525666688, 12.85845953116340891770256258454