L(s) = 1 | + (0.870 + 0.491i)2-s + (−0.564 − 0.825i)3-s + (0.516 + 0.856i)4-s + (0.362 + 0.931i)5-s + (−0.0855 − 0.996i)6-s + (−0.985 − 0.170i)7-s + (0.0285 + 0.999i)8-s + (−0.362 + 0.931i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.466 + 0.884i)13-s + (−0.774 − 0.633i)14-s + (0.564 − 0.825i)15-s + (−0.466 + 0.884i)16-s + (0.0855 + 0.996i)17-s + (−0.774 + 0.633i)18-s + ⋯ |
L(s) = 1 | + (0.870 + 0.491i)2-s + (−0.564 − 0.825i)3-s + (0.516 + 0.856i)4-s + (0.362 + 0.931i)5-s + (−0.0855 − 0.996i)6-s + (−0.985 − 0.170i)7-s + (0.0285 + 0.999i)8-s + (−0.362 + 0.931i)9-s + (−0.142 + 0.989i)10-s + (0.415 − 0.909i)12-s + (0.466 + 0.884i)13-s + (−0.774 − 0.633i)14-s + (0.564 − 0.825i)15-s + (−0.466 + 0.884i)16-s + (0.0855 + 0.996i)17-s + (−0.774 + 0.633i)18-s + ⋯ |
Λ(s)=(=(253s/2ΓR(s)L(s)(−0.0706+0.997i)Λ(1−s)
Λ(s)=(=(253s/2ΓR(s)L(s)(−0.0706+0.997i)Λ(1−s)
Degree: |
1 |
Conductor: |
253
= 11⋅23
|
Sign: |
−0.0706+0.997i
|
Analytic conductor: |
1.17492 |
Root analytic conductor: |
1.17492 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ253(182,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 253, (0: ), −0.0706+0.997i)
|
Particular Values
L(21) |
≈ |
0.9939225019+1.066779951i |
L(21) |
≈ |
0.9939225019+1.066779951i |
L(1) |
≈ |
1.209879006+0.5361712608i |
L(1) |
≈ |
1.209879006+0.5361712608i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 23 | 1 |
good | 2 | 1+(0.870+0.491i)T |
| 3 | 1+(−0.564−0.825i)T |
| 5 | 1+(0.362+0.931i)T |
| 7 | 1+(−0.985−0.170i)T |
| 13 | 1+(0.466+0.884i)T |
| 17 | 1+(0.0855+0.996i)T |
| 19 | 1+(−0.921−0.389i)T |
| 29 | 1+(0.921−0.389i)T |
| 31 | 1+(0.941−0.336i)T |
| 37 | 1+(0.254+0.967i)T |
| 41 | 1+(0.254−0.967i)T |
| 43 | 1+(−0.959+0.281i)T |
| 47 | 1+(0.309+0.951i)T |
| 53 | 1+(−0.696−0.717i)T |
| 59 | 1+(0.897+0.441i)T |
| 61 | 1+(0.610+0.791i)T |
| 67 | 1+(−0.415−0.909i)T |
| 71 | 1+(0.198−0.980i)T |
| 73 | 1+(−0.516−0.856i)T |
| 79 | 1+(−0.466−0.884i)T |
| 83 | 1+(−0.998−0.0570i)T |
| 89 | 1+(0.959−0.281i)T |
| 97 | 1+(0.998−0.0570i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.38407574775988623296284693428, −24.92019495405916773959329017966, −23.379258513372197129377679072939, −23.0596507302115314862249630118, −21.981433997040744536110497096733, −21.293900719030802553751081070216, −20.44329910121598681634103888370, −19.75343384932081827442003300362, −18.389127621646396237461122686100, −17.13322548365313620110009438703, −16.03830032268651305818554219343, −15.71423219035535636546573819281, −14.40398903139419107276187406554, −13.20325548497035445565249405013, −12.52187493815672853948716268034, −11.62637988683204896478239066129, −10.359541740733364665631427103638, −9.78123710788990933986178756980, −8.67092272259969672610998890969, −6.591517054820916737739855024590, −5.73917074036551961454996373402, −4.91663079376359819285252093692, −3.84972495129443649033213237356, −2.72894730448210457026703942633, −0.82410819818868308893590772090,
2.00512952740288485365269308959, 3.1185784364174613968735461338, 4.43820231037213787262799861971, 6.14941602828408900363805960775, 6.3202007833951404592793558357, 7.24315496637620115615912975977, 8.49636342613545725553045951099, 10.23932283530466005178282130396, 11.2191352832546855957356562908, 12.19420612056908515743903262226, 13.22723352783229743434899543605, 13.74480828591262742965269161574, 14.84590080085053847944254779518, 15.9406879163300474525683410274, 16.94470339341607643198102576472, 17.63101668408288283680042943359, 18.918018164475072118244668997190, 19.49633997346248308062359626830, 21.13418796541913128190372131548, 21.99051198742459952095239541636, 22.697596215841125246307340315735, 23.45099695557061223089950491794, 24.11527733706495389057669889875, 25.52611294622334174144577355114, 25.69255289110213704175218929724