L(s) = 1 | + (−1.35 − 0.784i)3-s + (38.3 − 22.1i)5-s + (42.3 + 24.5i)7-s + (−39.2 − 68.0i)9-s + (−11.7 + 20.3i)11-s + 136. i·13-s − 69.4·15-s + (−227. − 131. i)17-s + (−387. + 223. i)19-s + (−38.3 − 66.6i)21-s + (374. + 648. i)23-s + (667. − 1.15e3i)25-s + 250. i·27-s + 406.·29-s + (−584. − 337. i)31-s + ⋯ |
L(s) = 1 | + (−0.151 − 0.0871i)3-s + (1.53 − 0.885i)5-s + (0.865 + 0.501i)7-s + (−0.484 − 0.839i)9-s + (−0.0971 + 0.168i)11-s + 0.806i·13-s − 0.308·15-s + (−0.786 − 0.454i)17-s + (−1.07 + 0.619i)19-s + (−0.0869 − 0.151i)21-s + (0.708 + 1.22i)23-s + (1.06 − 1.84i)25-s + 0.343i·27-s + 0.483·29-s + (−0.607 − 0.350i)31-s + ⋯ |
Λ(s)=(=(28s/2ΓC(s)L(s)(0.921+0.387i)Λ(5−s)
Λ(s)=(=(28s/2ΓC(s+2)L(s)(0.921+0.387i)Λ(1−s)
Degree: |
2 |
Conductor: |
28
= 22⋅7
|
Sign: |
0.921+0.387i
|
Analytic conductor: |
2.89435 |
Root analytic conductor: |
1.70128 |
Motivic weight: |
4 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ28(17,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 28, ( :2), 0.921+0.387i)
|
Particular Values
L(25) |
≈ |
1.55340−0.313612i |
L(21) |
≈ |
1.55340−0.313612i |
L(3) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 7 | 1+(−42.3−24.5i)T |
good | 3 | 1+(1.35+0.784i)T+(40.5+70.1i)T2 |
| 5 | 1+(−38.3+22.1i)T+(312.5−541.i)T2 |
| 11 | 1+(11.7−20.3i)T+(−7.32e3−1.26e4i)T2 |
| 13 | 1−136.iT−2.85e4T2 |
| 17 | 1+(227.+131.i)T+(4.17e4+7.23e4i)T2 |
| 19 | 1+(387.−223.i)T+(6.51e4−1.12e5i)T2 |
| 23 | 1+(−374.−648.i)T+(−1.39e5+2.42e5i)T2 |
| 29 | 1−406.T+7.07e5T2 |
| 31 | 1+(584.+337.i)T+(4.61e5+7.99e5i)T2 |
| 37 | 1+(372.+645.i)T+(−9.37e5+1.62e6i)T2 |
| 41 | 1−2.47e3iT−2.82e6T2 |
| 43 | 1+2.63e3T+3.41e6T2 |
| 47 | 1+(579.−334.i)T+(2.43e6−4.22e6i)T2 |
| 53 | 1+(−1.01e3+1.75e3i)T+(−3.94e6−6.83e6i)T2 |
| 59 | 1+(1.01e3+585.i)T+(6.05e6+1.04e7i)T2 |
| 61 | 1+(2.02e3−1.16e3i)T+(6.92e6−1.19e7i)T2 |
| 67 | 1+(−3.77e3+6.54e3i)T+(−1.00e7−1.74e7i)T2 |
| 71 | 1−7.57e3T+2.54e7T2 |
| 73 | 1+(−3.28e3−1.89e3i)T+(1.41e7+2.45e7i)T2 |
| 79 | 1+(3.89e3+6.75e3i)T+(−1.94e7+3.37e7i)T2 |
| 83 | 1−2.18e3iT−4.74e7T2 |
| 89 | 1+(−8.05e3+4.64e3i)T+(3.13e7−5.43e7i)T2 |
| 97 | 1+9.98e3iT−8.85e7T2 |
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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
−16.74446766843733654455122525384, −15.00292285037885964820613862467, −13.86697844049336044157045620747, −12.68642173916509162703112111361, −11.38606255032621461100265094830, −9.558582809921076026651889991812, −8.674760391305448618824857250066, −6.30597490567660162390187619599, −5.00667049790379191886197397047, −1.80689940933409746069633287096,
2.31532916365648205235693675108, 5.16750799135441965390962438282, 6.67792386087337301726429095801, 8.540182932206538752322731057512, 10.49092140934562258115353080819, 10.85936506646538165828397582482, 13.14126412476617304548406991666, 14.04691261900454492100049185574, 15.03647174950760246300832044207, 16.99277902597652870338454631373