L(s) = 1 | + (−0.734 − 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (−0.329 + 1.01i)8-s + (0.987 + 0.156i)11-s + (−0.863 − 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s − 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (−0.0966 − 0.297i)29-s + 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (−0.0274 − 0.173i)44-s + ⋯ |
L(s) = 1 | + (−0.734 − 0.533i)2-s + (−0.0542 − 0.166i)4-s + (0.951 − 0.309i)7-s + (−0.329 + 1.01i)8-s + (0.987 + 0.156i)11-s + (−0.863 − 0.280i)14-s + (0.642 − 0.466i)16-s + (−0.642 − 0.642i)22-s − 1.78i·23-s + (−0.309 + 0.951i)25-s + (−0.103 − 0.142i)28-s + (−0.0966 − 0.297i)29-s + 0.346·32-s + (−0.587 − 1.80i)37-s + 1.61i·43-s + (−0.0274 − 0.173i)44-s + ⋯ |
Λ(s)=(=(693s/2ΓC(s)L(s)(0.402+0.915i)Λ(1−s)
Λ(s)=(=(693s/2ΓC(s)L(s)(0.402+0.915i)Λ(1−s)
Degree: |
2 |
Conductor: |
693
= 32⋅7⋅11
|
Sign: |
0.402+0.915i
|
Analytic conductor: |
0.345852 |
Root analytic conductor: |
0.588091 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ693(62,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 693, ( :0), 0.402+0.915i)
|
Particular Values
L(21) |
≈ |
0.7061868735 |
L(21) |
≈ |
0.7061868735 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(−0.951+0.309i)T |
| 11 | 1+(−0.987−0.156i)T |
good | 2 | 1+(0.734+0.533i)T+(0.309+0.951i)T2 |
| 5 | 1+(0.309−0.951i)T2 |
| 13 | 1+(0.309+0.951i)T2 |
| 17 | 1+(−0.309+0.951i)T2 |
| 19 | 1+(−0.809−0.587i)T2 |
| 23 | 1+1.78iT−T2 |
| 29 | 1+(0.0966+0.297i)T+(−0.809+0.587i)T2 |
| 31 | 1+(−0.309−0.951i)T2 |
| 37 | 1+(0.587+1.80i)T+(−0.809+0.587i)T2 |
| 41 | 1+(0.809+0.587i)T2 |
| 43 | 1−1.61iT−T2 |
| 47 | 1+(−0.809−0.587i)T2 |
| 53 | 1+(−0.183+0.253i)T+(−0.309−0.951i)T2 |
| 59 | 1+(−0.809+0.587i)T2 |
| 61 | 1+(0.309−0.951i)T2 |
| 67 | 1+1.61T+T2 |
| 71 | 1+(−1.16−1.59i)T+(−0.309+0.951i)T2 |
| 73 | 1+(−0.809+0.587i)T2 |
| 79 | 1+(0.690−0.951i)T+(−0.309−0.951i)T2 |
| 83 | 1+(−0.309+0.951i)T2 |
| 89 | 1+T2 |
| 97 | 1+(−0.309−0.951i)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
−10.57349426378619545675900608310, −9.653174689899536106897026409875, −8.905522486581032259786156313159, −8.196570808172035148461442265304, −7.18288744374796686724519578318, −6.04606971751183770991396506099, −4.98461130327721379562703744967, −4.00680523639470954762419679314, −2.35405747009839158525114193277, −1.23470822574348972553277143840,
1.56068324131744474829727750197, 3.33878064382717655120915340626, 4.38839005943541075756168089498, 5.61202361112755444232928647323, 6.64546497063641141844514118203, 7.50340724305552595658435391941, 8.284580128500657370635364216275, 8.949765729037654034744893271477, 9.691169460784368002436702431350, 10.69672024028364157074166705647