L(s) = 1 | + (−0.642 + 0.766i)3-s + (0.866 − 0.5i)7-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (0.866 + 0.5i)27-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.342 − 0.939i)33-s − i·37-s + 39-s + (0.766 + 0.642i)41-s + ⋯ |
L(s) = 1 | + (−0.642 + 0.766i)3-s + (0.866 − 0.5i)7-s + (−0.173 − 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.642 − 0.766i)13-s + (−0.984 − 0.173i)17-s + (−0.173 + 0.984i)21-s + (−0.342 + 0.939i)23-s + (0.866 + 0.5i)27-s + (0.173 + 0.984i)29-s + (0.5 + 0.866i)31-s + (−0.342 − 0.939i)33-s − i·37-s + 39-s + (0.766 + 0.642i)41-s + ⋯ |
Λ(s)=(=(760s/2ΓR(s)L(s)(−0.460+0.887i)Λ(1−s)
Λ(s)=(=(760s/2ΓR(s)L(s)(−0.460+0.887i)Λ(1−s)
Degree: |
1 |
Conductor: |
760
= 23⋅5⋅19
|
Sign: |
−0.460+0.887i
|
Analytic conductor: |
3.52942 |
Root analytic conductor: |
3.52942 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ760(443,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 760, (0: ), −0.460+0.887i)
|
Particular Values
L(21) |
≈ |
0.4134321972+0.6801680654i |
L(21) |
≈ |
0.4134321972+0.6801680654i |
L(1) |
≈ |
0.7576383149+0.2568738252i |
L(1) |
≈ |
0.7576383149+0.2568738252i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 19 | 1 |
good | 3 | 1+(−0.642+0.766i)T |
| 7 | 1+(0.866−0.5i)T |
| 11 | 1+(−0.5+0.866i)T |
| 13 | 1+(−0.642−0.766i)T |
| 17 | 1+(−0.984−0.173i)T |
| 23 | 1+(−0.342+0.939i)T |
| 29 | 1+(0.173+0.984i)T |
| 31 | 1+(0.5+0.866i)T |
| 37 | 1−iT |
| 41 | 1+(0.766+0.642i)T |
| 43 | 1+(−0.342−0.939i)T |
| 47 | 1+(−0.984+0.173i)T |
| 53 | 1+(−0.342+0.939i)T |
| 59 | 1+(−0.173+0.984i)T |
| 61 | 1+(0.939+0.342i)T |
| 67 | 1+(0.984−0.173i)T |
| 71 | 1+(0.939−0.342i)T |
| 73 | 1+(−0.642+0.766i)T |
| 79 | 1+(0.766+0.642i)T |
| 83 | 1+(0.866−0.5i)T |
| 89 | 1+(−0.766+0.642i)T |
| 97 | 1+(−0.984−0.173i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.11575928865536331248729699458, −21.50296932871845535942479973730, −20.643134015466349221856039416012, −19.40288507219837781355952870230, −18.925434423561648488224372732678, −18.028876946500708767150003255516, −17.48040863199519879441210894146, −16.565093522072225380269111532106, −15.79054768439123103652835125394, −14.65594306525382571132853564046, −13.90532916058386091546218235635, −13.04246028445459096383920598733, −12.19177652388555389920768253825, −11.33869627622453243329802513084, −10.938767328691461962834889337511, −9.63986485877194683240462844117, −8.415452883921464585989392769229, −7.94155362370984090045339831714, −6.77430553199828026753483071637, −6.0227855306820711035423358429, −5.09383655291551403442024802420, −4.27306089886836964473669721795, −2.510839547839773874658873538707, −1.93814389626036392091066916048, −0.422841236686348237670871056774,
1.25740245021139799766955510478, 2.63899648986183781940425559505, 3.879174047194732012323292715924, 4.88000353738630575968331980856, 5.205899717535006215371456801863, 6.58153631464902577322243342097, 7.4564217475766663131675173838, 8.43208066452110896280895559490, 9.5720584549426314261438181822, 10.30714209828863551180844981593, 10.9682341508132892374608389629, 11.83730271295928907469290818967, 12.66127979603714317214080918324, 13.73100513448362566474021602362, 14.77404004103721040189053146496, 15.33867572806124537779954877865, 16.1179920850621920738490634257, 17.21683597966165589515765773194, 17.66971060457041640929557639546, 18.241082294841189741087177253212, 19.85275461245167187826454502605, 20.2589131499033481853837983450, 21.13776478749200477790896468250, 21.86400520391079985495350722780, 22.648150593608323453870831503604