L(s) = 1 | + (1.02 + 1.28i)2-s + (−0.900 + 0.433i)3-s + (−0.155 + 0.679i)4-s + (−0.900 + 0.433i)5-s + (−1.47 − 0.712i)6-s + (−0.516 − 2.59i)7-s + (1.92 − 0.928i)8-s + (0.623 − 0.781i)9-s + (−1.47 − 0.712i)10-s + (−2.66 − 3.34i)11-s + (−0.155 − 0.679i)12-s + (−0.474 − 0.594i)13-s + (2.80 − 3.32i)14-s + (0.623 − 0.781i)15-s + (4.42 + 2.12i)16-s + (−0.265 − 1.16i)17-s + ⋯ |
L(s) = 1 | + (0.723 + 0.907i)2-s + (−0.520 + 0.250i)3-s + (−0.0775 + 0.339i)4-s + (−0.402 + 0.194i)5-s + (−0.604 − 0.290i)6-s + (−0.195 − 0.980i)7-s + (0.681 − 0.328i)8-s + (0.207 − 0.260i)9-s + (−0.467 − 0.225i)10-s + (−0.804 − 1.00i)11-s + (−0.0447 − 0.196i)12-s + (−0.131 − 0.164i)13-s + (0.748 − 0.887i)14-s + (0.160 − 0.201i)15-s + (1.10 + 0.532i)16-s + (−0.0645 − 0.282i)17-s + ⋯ |
Λ(s)=(=(735s/2ΓC(s)L(s)(0.930+0.367i)Λ(2−s)
Λ(s)=(=(735s/2ΓC(s+1/2)L(s)(0.930+0.367i)Λ(1−s)
Degree: |
2 |
Conductor: |
735
= 3⋅5⋅72
|
Sign: |
0.930+0.367i
|
Analytic conductor: |
5.86900 |
Root analytic conductor: |
2.42260 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ735(106,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 735, ( :1/2), 0.930+0.367i)
|
Particular Values
L(1) |
≈ |
1.49151−0.283815i |
L(21) |
≈ |
1.49151−0.283815i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.900−0.433i)T |
| 5 | 1+(0.900−0.433i)T |
| 7 | 1+(0.516+2.59i)T |
good | 2 | 1+(−1.02−1.28i)T+(−0.445+1.94i)T2 |
| 11 | 1+(2.66+3.34i)T+(−2.44+10.7i)T2 |
| 13 | 1+(0.474+0.594i)T+(−2.89+12.6i)T2 |
| 17 | 1+(0.265+1.16i)T+(−15.3+7.37i)T2 |
| 19 | 1−1.64T+19T2 |
| 23 | 1+(−1.20+5.28i)T+(−20.7−9.97i)T2 |
| 29 | 1+(2.07+9.10i)T+(−26.1+12.5i)T2 |
| 31 | 1−4.21T+31T2 |
| 37 | 1+(−1.27−5.58i)T+(−33.3+16.0i)T2 |
| 41 | 1+(6.12−2.94i)T+(25.5−32.0i)T2 |
| 43 | 1+(−4.61−2.22i)T+(26.8+33.6i)T2 |
| 47 | 1+(−4.12−5.17i)T+(−10.4+45.8i)T2 |
| 53 | 1+(0.137−0.604i)T+(−47.7−22.9i)T2 |
| 59 | 1+(3.28+1.58i)T+(36.7+46.1i)T2 |
| 61 | 1+(−0.132−0.579i)T+(−54.9+26.4i)T2 |
| 67 | 1+2.44T+67T2 |
| 71 | 1+(−2.71+11.8i)T+(−63.9−30.8i)T2 |
| 73 | 1+(−1.62+2.03i)T+(−16.2−71.1i)T2 |
| 79 | 1+1.93T+79T2 |
| 83 | 1+(−7.50+9.41i)T+(−18.4−80.9i)T2 |
| 89 | 1+(5.60−7.02i)T+(−19.8−86.7i)T2 |
| 97 | 1+4.69T+97T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.49297557838995596955261652841, −9.683432841369340874669379269444, −8.158497707907155539516918796523, −7.57980679602527753647332001321, −6.58418925620560036690767947093, −5.98361294345708480629158437288, −4.90268530011738926493533171992, −4.22858305530169705941475732070, −3.07274792972762016141991539880, −0.67527739908887703062810409061,
1.71354571167351888299979048064, 2.74977503059724128707715059685, 3.89334553473021982861660667135, 5.05283292106721538602536246322, 5.49061928013307745055473070909, 6.99674384643241970313343665476, 7.70938291364980036428582281760, 8.795884632818370160407805361549, 9.857079537134266912086844642556, 10.71923057084321514191894741625