L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + (−0.965 + 0.258i)6-s + (−0.258 − 0.965i)7-s + 0.999·8-s + 1.00i·9-s + (0.707 − 0.707i)10-s + (0.866 + 0.5i)11-s + (0.258 − 0.965i)12-s + (0.965 + 0.258i)13-s + (0.965 + 0.258i)14-s + (−0.500 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)2-s + (0.707 + 0.707i)3-s + (−0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + (−0.965 + 0.258i)6-s + (−0.258 − 0.965i)7-s + 0.999·8-s + 1.00i·9-s + (0.707 − 0.707i)10-s + (0.866 + 0.5i)11-s + (0.258 − 0.965i)12-s + (0.965 + 0.258i)13-s + (0.965 + 0.258i)14-s + (−0.500 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (0.707 + 0.707i)17-s + ⋯ |
Λ(s)=(=(1260s/2ΓC(s)L(s)(0.203−0.979i)Λ(1−s)
Λ(s)=(=(1260s/2ΓC(s)L(s)(0.203−0.979i)Λ(1−s)
Degree: |
2 |
Conductor: |
1260
= 22⋅32⋅5⋅7
|
Sign: |
0.203−0.979i
|
Analytic conductor: |
0.628821 |
Root analytic conductor: |
0.792982 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1260(223,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1260, ( :0), 0.203−0.979i)
|
Particular Values
L(21) |
≈ |
0.9124862028 |
L(21) |
≈ |
0.9124862028 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.5−0.866i)T |
| 3 | 1+(−0.707−0.707i)T |
| 5 | 1+(0.965+0.258i)T |
| 7 | 1+(0.258+0.965i)T |
good | 11 | 1+(−0.866−0.5i)T+(0.5+0.866i)T2 |
| 13 | 1+(−0.965−0.258i)T+(0.866+0.5i)T2 |
| 17 | 1+(−0.707−0.707i)T+iT2 |
| 19 | 1+1.41iT−T2 |
| 23 | 1+(0.866+0.5i)T2 |
| 29 | 1+(0.5+0.866i)T2 |
| 31 | 1+(−0.5+0.866i)T2 |
| 37 | 1−iT2 |
| 41 | 1+(0.5−0.866i)T2 |
| 43 | 1+(−0.366−1.36i)T+(−0.866+0.5i)T2 |
| 47 | 1+(−0.258−0.965i)T+(−0.866+0.5i)T2 |
| 53 | 1+(−1−i)T+iT2 |
| 59 | 1+(0.5−0.866i)T2 |
| 61 | 1+(1.22+0.707i)T+(0.5+0.866i)T2 |
| 67 | 1+(−0.866−0.5i)T2 |
| 71 | 1+iT−T2 |
| 73 | 1+(−0.707+0.707i)T−iT2 |
| 79 | 1+(−0.5+0.866i)T+(−0.5−0.866i)T2 |
| 83 | 1+(0.965−0.258i)T+(0.866−0.5i)T2 |
| 89 | 1+T2 |
| 97 | 1+(−0.965+0.258i)T+(0.866−0.5i)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
−9.722348352656257875525056198489, −9.165609244161786881390500377572, −8.437828634573039179718705842854, −7.68217021193605930972216487332, −7.05921947660831919266869931870, −6.08945604046071282957409914592, −4.65624388655411653268196830212, −4.25016965665686481607256709995, −3.33526921864234716435143771297, −1.29708298378638582106939934705,
1.13191424589445376553934487090, 2.45135935143368542288086108791, 3.50196319214261299705860292629, 3.81627334095731822982170926284, 5.57958386770808675404901404909, 6.66509403237882870854163402385, 7.55810724212145294074289564784, 8.384290660514656024378683413879, 8.706259701765121751738961877038, 9.567237433432570300003375953186