L(s) = 1 | + (1.78 − 0.859i)2-s + (−0.623 − 0.781i)3-s + (1.19 − 1.50i)4-s + (−1.09 + 0.528i)5-s + (−1.78 − 0.859i)6-s + (0.245 + 0.307i)7-s + (−0.0337 + 0.147i)8-s + (−0.222 + 0.974i)9-s + (−1.50 + 1.88i)10-s + (0.554 + 2.43i)11-s − 1.92·12-s + (−0.735 − 3.22i)13-s + (0.701 + 0.337i)14-s + (1.09 + 0.528i)15-s + (0.922 + 4.04i)16-s − 4.30·17-s + ⋯ |
L(s) = 1 | + (1.26 − 0.607i)2-s + (−0.359 − 0.451i)3-s + (0.599 − 0.751i)4-s + (−0.491 + 0.236i)5-s + (−0.728 − 0.350i)6-s + (0.0926 + 0.116i)7-s + (−0.0119 + 0.0522i)8-s + (−0.0741 + 0.324i)9-s + (−0.476 + 0.596i)10-s + (0.167 + 0.733i)11-s − 0.555·12-s + (−0.203 − 0.893i)13-s + (0.187 + 0.0903i)14-s + (0.283 + 0.136i)15-s + (0.230 + 1.01i)16-s − 1.04·17-s + ⋯ |
Λ(s)=(=(87s/2ΓC(s)L(s)(0.648+0.760i)Λ(2−s)
Λ(s)=(=(87s/2ΓC(s+1/2)L(s)(0.648+0.760i)Λ(1−s)
Degree: |
2 |
Conductor: |
87
= 3⋅29
|
Sign: |
0.648+0.760i
|
Analytic conductor: |
0.694698 |
Root analytic conductor: |
0.833485 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ87(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 87, ( :1/2), 0.648+0.760i)
|
Particular Values
L(1) |
≈ |
1.31901−0.608749i |
L(21) |
≈ |
1.31901−0.608749i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.623+0.781i)T |
| 29 | 1+(1.86+5.05i)T |
good | 2 | 1+(−1.78+0.859i)T+(1.24−1.56i)T2 |
| 5 | 1+(1.09−0.528i)T+(3.11−3.90i)T2 |
| 7 | 1+(−0.245−0.307i)T+(−1.55+6.82i)T2 |
| 11 | 1+(−0.554−2.43i)T+(−9.91+4.77i)T2 |
| 13 | 1+(0.735+3.22i)T+(−11.7+5.64i)T2 |
| 17 | 1+4.30T+17T2 |
| 19 | 1+(−3.34+4.19i)T+(−4.22−18.5i)T2 |
| 23 | 1+(2.70+1.30i)T+(14.3+17.9i)T2 |
| 31 | 1+(−9.01+4.34i)T+(19.3−24.2i)T2 |
| 37 | 1+(0.812−3.55i)T+(−33.3−16.0i)T2 |
| 41 | 1+7.82T+41T2 |
| 43 | 1+(2.11+1.01i)T+(26.8+33.6i)T2 |
| 47 | 1+(−1.90−8.32i)T+(−42.3+20.3i)T2 |
| 53 | 1+(−9.83+4.73i)T+(33.0−41.4i)T2 |
| 59 | 1−10.2T+59T2 |
| 61 | 1+(−2.32−2.91i)T+(−13.5+59.4i)T2 |
| 67 | 1+(1.14−5.01i)T+(−60.3−29.0i)T2 |
| 71 | 1+(−1.80−7.88i)T+(−63.9+30.8i)T2 |
| 73 | 1+(7.88+3.79i)T+(45.5+57.0i)T2 |
| 79 | 1+(1.62−7.11i)T+(−71.1−34.2i)T2 |
| 83 | 1+(2.33−2.93i)T+(−18.4−80.9i)T2 |
| 89 | 1+(9.38−4.51i)T+(55.4−69.5i)T2 |
| 97 | 1+(10.1−12.6i)T+(−21.5−94.5i)T2 |
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
−13.63692981013337820844174188839, −13.06333603176451627867364086828, −11.84577863036101994033090550882, −11.46898762746772847264017701163, −10.08377564760850675841894361234, −8.223933758845294494959907695022, −6.85902930013229628673272700306, −5.42641800393009191105941480260, −4.21712110329007367991069909983, −2.54215679615060119698579005786,
3.66740160253107730561586249301, 4.67441406496771689143352280681, 5.90853433365689050427506205142, 7.05101291507953887870061642382, 8.623092770098183662905856474212, 10.08496482764488674377084844213, 11.58396300389461956135187849472, 12.20746569336512079093491929639, 13.61813415860792534176184067111, 14.23330579335308102454371215832