L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.382 + 0.923i)3-s + i·4-s + (0.923 − 0.382i)6-s + (−0.923 + 0.382i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)12-s − 13-s + (0.923 + 0.382i)14-s − 16-s + i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (0.923 + 0.382i)22-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + (−0.382 + 0.923i)3-s + i·4-s + (0.923 − 0.382i)6-s + (−0.923 + 0.382i)7-s + (0.707 − 0.707i)8-s + (−0.707 − 0.707i)9-s + (−0.923 + 0.382i)11-s + (−0.923 − 0.382i)12-s − 13-s + (0.923 + 0.382i)14-s − 16-s + i·18-s + (−0.707 + 0.707i)19-s − i·21-s + (0.923 + 0.382i)22-s + ⋯ |
Λ(s)=(=(85s/2ΓR(s)L(s)(−0.724+0.688i)Λ(1−s)
Λ(s)=(=(85s/2ΓR(s)L(s)(−0.724+0.688i)Λ(1−s)
Degree: |
1 |
Conductor: |
85
= 5⋅17
|
Sign: |
−0.724+0.688i
|
Analytic conductor: |
0.394738 |
Root analytic conductor: |
0.394738 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ85(3,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 85, (0: ), −0.724+0.688i)
|
Particular Values
L(21) |
≈ |
0.08928106553+0.2235191158i |
L(21) |
≈ |
0.08928106553+0.2235191158i |
L(1) |
≈ |
0.4294688579+0.1020763067i |
L(1) |
≈ |
0.4294688579+0.1020763067i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1 |
good | 2 | 1+(−0.707−0.707i)T |
| 3 | 1+(−0.382+0.923i)T |
| 7 | 1+(−0.923+0.382i)T |
| 11 | 1+(−0.923+0.382i)T |
| 13 | 1−T |
| 19 | 1+(−0.707+0.707i)T |
| 23 | 1+(0.382+0.923i)T |
| 29 | 1+(−0.382+0.923i)T |
| 31 | 1+(−0.923−0.382i)T |
| 37 | 1+(0.382−0.923i)T |
| 41 | 1+(−0.382−0.923i)T |
| 43 | 1+(−0.707+0.707i)T |
| 47 | 1+T |
| 53 | 1+(0.707+0.707i)T |
| 59 | 1+(0.707+0.707i)T |
| 61 | 1+(0.382+0.923i)T |
| 67 | 1+iT |
| 71 | 1+(0.923+0.382i)T |
| 73 | 1+(−0.923−0.382i)T |
| 79 | 1+(0.923−0.382i)T |
| 83 | 1+(−0.707−0.707i)T |
| 89 | 1+iT |
| 97 | 1+(−0.923−0.382i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−30.02943377449248199554825787503, −29.028106680099927868205628653130, −28.49737564867443819906327725773, −26.98961235354758927103751802464, −26.03222196707254168069712146046, −25.081445234739382006271453557, −24.01033708281403746228697970690, −23.3002627955444567759545672829, −22.17126986148278219061676782728, −20.13479190047071827161064413313, −19.1867255434862138607227569429, −18.43100124549418401493144166779, −17.16794331678531674442694160403, −16.489040350333895067217739805935, −15.12394455564625498691665335873, −13.68599868644918342182133685245, −12.73027214438804170281741796777, −11.06874636715782241608524832818, −9.94953776400973228622195903742, −8.43943994609804439044077628646, −7.26698260075404946332182931229, −6.40218057799490864212948522330, −5.09060132704338165328197960169, −2.43525560267084227011274805788, −0.31808636057728767729118627245,
2.54528249262928664135182877806, 3.85705240619494951152021901707, 5.45789470828319650454728115623, 7.27328334579718767942258940877, 8.9231071825530929938929691276, 9.85596342223925887134491441122, 10.68214302782606843679258552689, 12.04824537621092211911797925209, 12.97225653484040312459631900597, 14.97990612886508400396814252090, 16.11950886094157234681238326527, 16.97898274557538364724628110947, 18.15871832681635481511565606828, 19.359042920622371440340294804285, 20.40612520185525379009281912081, 21.485476647705395588432827675047, 22.23516912783899901982705972547, 23.344105792424552006714056172924, 25.333580252382846115729520199, 26.10807539844722781299942126105, 27.11310135232674559400304153313, 28.01238570633121481938327570702, 28.984276359289998788994577742464, 29.55826032559495271943083024882, 31.40134044194414453549314904721