L(s) = 1 | + (−0.5 + 0.866i)2-s + (3.70 − 6.42i)3-s + (3.5 + 6.06i)4-s + (−0.208 + 0.360i)5-s + (3.70 + 6.42i)6-s − 14.4·7-s − 15·8-s + (−14 − 24.2i)9-s + (−0.208 − 0.360i)10-s − 37.9·11-s + 51.9·12-s + (36.6 + 63.5i)13-s + (7.20 − 12.4i)14-s + (1.54 + 2.67i)15-s + (−20.5 + 35.5i)16-s + (57.6 − 99.8i)17-s + ⋯ |
L(s) = 1 | + (−0.176 + 0.306i)2-s + (0.713 − 1.23i)3-s + (0.437 + 0.757i)4-s + (−0.0186 + 0.0322i)5-s + (0.252 + 0.437i)6-s − 0.778·7-s − 0.662·8-s + (−0.518 − 0.898i)9-s + (−0.00658 − 0.0113i)10-s − 1.03·11-s + 1.24·12-s + (0.782 + 1.35i)13-s + (0.137 − 0.238i)14-s + (0.0265 + 0.0460i)15-s + (−0.320 + 0.554i)16-s + (0.822 − 1.42i)17-s + ⋯ |
Λ(s)=(=(19s/2ΓC(s)L(s)(0.992+0.120i)Λ(4−s)
Λ(s)=(=(19s/2ΓC(s+3/2)L(s)(0.992+0.120i)Λ(1−s)
Degree: |
2 |
Conductor: |
19
|
Sign: |
0.992+0.120i
|
Analytic conductor: |
1.12103 |
Root analytic conductor: |
1.05879 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ19(11,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 19, ( :3/2), 0.992+0.120i)
|
Particular Values
L(2) |
≈ |
1.16326−0.0704157i |
L(21) |
≈ |
1.16326−0.0704157i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 19 | 1+(15.6+81.3i)T |
good | 2 | 1+(0.5−0.866i)T+(−4−6.92i)T2 |
| 3 | 1+(−3.70+6.42i)T+(−13.5−23.3i)T2 |
| 5 | 1+(0.208−0.360i)T+(−62.5−108.i)T2 |
| 7 | 1+14.4T+343T2 |
| 11 | 1+37.9T+1.33e3T2 |
| 13 | 1+(−36.6−63.5i)T+(−1.09e3+1.90e3i)T2 |
| 17 | 1+(−57.6+99.8i)T+(−2.45e3−4.25e3i)T2 |
| 23 | 1+(2.54+4.40i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(81.4+141.i)T+(−1.21e4+2.11e4i)T2 |
| 31 | 1−81.0T+2.97e4T2 |
| 37 | 1−39.0T+5.06e4T2 |
| 41 | 1+(164.−285.i)T+(−3.44e4−5.96e4i)T2 |
| 43 | 1+(−88.7+153.i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1+(−24.1−41.7i)T+(−5.19e4+8.99e4i)T2 |
| 53 | 1+(−324.−562.i)T+(−7.44e4+1.28e5i)T2 |
| 59 | 1+(−20.2+34.9i)T+(−1.02e5−1.77e5i)T2 |
| 61 | 1+(244.+422.i)T+(−1.13e5+1.96e5i)T2 |
| 67 | 1+(48.6+84.1i)T+(−1.50e5+2.60e5i)T2 |
| 71 | 1+(−47.7+82.6i)T+(−1.78e5−3.09e5i)T2 |
| 73 | 1+(65.2−113.i)T+(−1.94e5−3.36e5i)T2 |
| 79 | 1+(169.−293.i)T+(−2.46e5−4.26e5i)T2 |
| 83 | 1+1.21e3T+5.71e5T2 |
| 89 | 1+(−357.−619.i)T+(−3.52e5+6.10e5i)T2 |
| 97 | 1+(93.4−161.i)T+(−4.56e5−7.90e5i)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
−18.31746274463354472166707796772, −16.70477352897153017797265422343, −15.61956858362561245054527816768, −13.74907908827573654150726878296, −12.92528042527716021954554939053, −11.59180350846452901703352689354, −9.095862318629453283956321728588, −7.66963732698634338869232858888, −6.67815208648970580659865166938, −2.81080177139999889406601552746,
3.28410990329869220790991015590, 5.73887525915786512917881324531, 8.437267932235750411788221689331, 10.17266619050044811358202967564, 10.45447838680352329623674941604, 12.76643311611987639263565177189, 14.58065532351422642765923631671, 15.47735260698737009456932652187, 16.29719430542360245890068302798, 18.38623280751933292867241115970