L(s) = 1 | − 1.46i·2-s + 0.114·3-s + 1.86·4-s + 2.23·5-s − 0.167i·6-s − 4.56i·7-s − 8.56i·8-s − 8.98·9-s − 3.26i·10-s + (5.89 + 9.28i)11-s + 0.213·12-s + 16.5i·13-s − 6.66·14-s + 0.256·15-s − 5.05·16-s − 17.2i·17-s + ⋯ |
L(s) = 1 | − 0.730i·2-s + 0.0381·3-s + 0.466·4-s + 0.447·5-s − 0.0278i·6-s − 0.651i·7-s − 1.07i·8-s − 0.998·9-s − 0.326i·10-s + (0.535 + 0.844i)11-s + 0.0178·12-s + 1.27i·13-s − 0.475·14-s + 0.0170·15-s − 0.316·16-s − 1.01i·17-s + ⋯ |
Λ(s)=(=(55s/2ΓC(s)L(s)(0.535+0.844i)Λ(3−s)
Λ(s)=(=(55s/2ΓC(s+1)L(s)(0.535+0.844i)Λ(1−s)
Degree: |
2 |
Conductor: |
55
= 5⋅11
|
Sign: |
0.535+0.844i
|
Analytic conductor: |
1.49864 |
Root analytic conductor: |
1.22419 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ55(21,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 55, ( :1), 0.535+0.844i)
|
Particular Values
L(23) |
≈ |
1.17395−0.645527i |
L(21) |
≈ |
1.17395−0.645527i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1−2.23T |
| 11 | 1+(−5.89−9.28i)T |
good | 2 | 1+1.46iT−4T2 |
| 3 | 1−0.114T+9T2 |
| 7 | 1+4.56iT−49T2 |
| 13 | 1−16.5iT−169T2 |
| 17 | 1+17.2iT−289T2 |
| 19 | 1−35.8iT−361T2 |
| 23 | 1+29.3T+529T2 |
| 29 | 1+8.51iT−841T2 |
| 31 | 1+26.3T+961T2 |
| 37 | 1−44.4T+1.36e3T2 |
| 41 | 1+52.2iT−1.68e3T2 |
| 43 | 1+6.77iT−1.84e3T2 |
| 47 | 1−15.0T+2.20e3T2 |
| 53 | 1+33.1T+2.80e3T2 |
| 59 | 1−51.5T+3.48e3T2 |
| 61 | 1+23.1iT−3.72e3T2 |
| 67 | 1+113.T+4.48e3T2 |
| 71 | 1−8.00T+5.04e3T2 |
| 73 | 1+32.5iT−5.32e3T2 |
| 79 | 1+52.0iT−6.24e3T2 |
| 83 | 1−43.3iT−6.88e3T2 |
| 89 | 1−73.8T+7.92e3T2 |
| 97 | 1−22.0T+9.40e3T2 |
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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
−14.57000515499014124995599328951, −13.84579412680982573285077653821, −12.26338763133091248962446924422, −11.56688315435591868077543148778, −10.28596601990578832311633424353, −9.327769931827861834775259070508, −7.43969702475584404203980853587, −6.12679230052008586798972981634, −3.93831125514143753622104466982, −1.99565778490526600045696685395,
2.75584136954852601515315071505, 5.54713345154897930472551426192, 6.27908360837472805645640476979, 8.015837218665088524393195647623, 8.966561144628718457845937824634, 10.77666589082309009433420756659, 11.70402536067062103735555902640, 13.18446145816026083388920711875, 14.46523059179520259584166715042, 15.20109975396267457968008651279