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+1)L(s)(0.144+0.989i)Λ(1−s)
Λ(s)=(=(760s/2ΓR(s+1)L(s)(0.144+0.989i)Λ(1−s)
Degree: |
1 |
Conductor: |
760
= 23⋅5⋅19
|
Sign: |
0.144+0.989i
|
Analytic conductor: |
81.6733 |
Root analytic conductor: |
81.6733 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ760(507,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 760, (1: ), 0.144+0.989i)
|
Particular Values
L(21) |
≈ |
0.7050572728+0.6094494265i |
L(21) |
≈ |
0.7050572728+0.6094494265i |
L(1) |
≈ |
0.6931103361+0.1969789036i |
L(1) |
≈ |
0.6931103361+0.1969789036i |
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
−22.00715143128472983606376136963, −21.50410165913331622882384928561, −20.25668469478385691558240175587, −19.285397133716598698035366402309, −18.97450412643259099976581479991, −18.03748346818140111562148906565, −17.050613571800287109129060093426, −16.474226496188999507738911267673, −15.86465687309798473772529979422, −14.384962208009346726123976441659, −13.744394354540920531580903982383, −12.90167773782433141550637788505, −12.23639674823704429009318133649, −11.29907438603118104489348299755, −10.5169935212600565224452851416, −9.597574864680680455839801741852, −8.50415636884229109013056425821, −7.32352043331475066342246215630, −6.958717420693916807684977315164, −5.78311924396985953338203277918, −5.16849115120727584489609864884, −3.73663690744993946731850964611, −2.7650811244468512912135899093, −1.45940890337953272252140861622, −0.3961051783918262991603893880,
0.61636821450009234788703199225, 2.403966318335426569445511815590, 3.31185027261953433915891907732, 4.41699116738224151582095968519, 5.33601453742702006417496241342, 6.02773611149079249305407929545, 7.06074140739109549116875860825, 8.12967487524772643984599276989, 9.3544305819126467453313565065, 9.96235049127422260692518607795, 10.53880101126422050178123154105, 11.7602069114106258593131549518, 12.45167626512931117757227803013, 13.06891492796902351219936621296, 14.5486171268066207308080809205, 15.2326862078895134575819858050, 15.79291414029997213711893382055, 16.800586038723304611840262628352, 17.30283084689924671593180346603, 18.37591727984166005954938931544, 19.033566252432319826723607693765, 20.270513244710874082406794222162, 20.72496511448246537766759477337, 21.77396255369341927171547313163, 22.43715874612859759405615120927