L(s) = 1 | + (−0.393 + 0.919i)2-s + (0.691 − 0.722i)3-s + (−0.691 − 0.722i)4-s + (0.753 + 0.657i)5-s + (0.393 + 0.919i)6-s + (−0.134 + 0.990i)7-s + (0.936 − 0.351i)8-s + (−0.0448 − 0.998i)9-s + (−0.900 + 0.433i)10-s − 12-s + (0.963 − 0.266i)13-s + (−0.858 − 0.512i)14-s + (0.995 − 0.0896i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + (0.936 + 0.351i)18-s + ⋯ |
L(s) = 1 | + (−0.393 + 0.919i)2-s + (0.691 − 0.722i)3-s + (−0.691 − 0.722i)4-s + (0.753 + 0.657i)5-s + (0.393 + 0.919i)6-s + (−0.134 + 0.990i)7-s + (0.936 − 0.351i)8-s + (−0.0448 − 0.998i)9-s + (−0.900 + 0.433i)10-s − 12-s + (0.963 − 0.266i)13-s + (−0.858 − 0.512i)14-s + (0.995 − 0.0896i)15-s + (−0.0448 + 0.998i)16-s + (−0.809 + 0.587i)17-s + (0.936 + 0.351i)18-s + ⋯ |
Λ(s)=(=(319s/2ΓR(s+1)L(s)(−0.282+0.959i)Λ(1−s)
Λ(s)=(=(319s/2ΓR(s+1)L(s)(−0.282+0.959i)Λ(1−s)
Degree: |
1 |
Conductor: |
319
= 11⋅29
|
Sign: |
−0.282+0.959i
|
Analytic conductor: |
34.2813 |
Root analytic conductor: |
34.2813 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ319(35,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 319, (1: ), −0.282+0.959i)
|
Particular Values
L(21) |
≈ |
1.215306675+1.623945674i |
L(21) |
≈ |
1.215306675+1.623945674i |
L(1) |
≈ |
1.075867298+0.5360408484i |
L(1) |
≈ |
1.075867298+0.5360408484i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1 |
| 29 | 1 |
good | 2 | 1+(−0.393+0.919i)T |
| 3 | 1+(0.691−0.722i)T |
| 5 | 1+(0.753+0.657i)T |
| 7 | 1+(−0.134+0.990i)T |
| 13 | 1+(0.963−0.266i)T |
| 17 | 1+(−0.809+0.587i)T |
| 19 | 1+(0.134+0.990i)T |
| 23 | 1+(−0.222+0.974i)T |
| 31 | 1+(0.393−0.919i)T |
| 37 | 1+(0.550−0.834i)T |
| 41 | 1+(0.309−0.951i)T |
| 43 | 1+(−0.222+0.974i)T |
| 47 | 1+(0.550+0.834i)T |
| 53 | 1+(−0.393+0.919i)T |
| 59 | 1+(0.309+0.951i)T |
| 61 | 1+(0.983−0.178i)T |
| 67 | 1+(0.623+0.781i)T |
| 71 | 1+(−0.0448+0.998i)T |
| 73 | 1+(−0.995+0.0896i)T |
| 79 | 1+(−0.0448−0.998i)T |
| 83 | 1+(−0.473+0.880i)T |
| 89 | 1+(0.222+0.974i)T |
| 97 | 1+(−0.983−0.178i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−24.98983206893273959687818388495, −23.70290817426899943913517458412, −22.487122322627253909036774291488, −21.71830075729900224643325372179, −20.83106546247139634880953049184, −20.30037747335539465279458735929, −19.71216743466135543717602593347, −18.45538857795089021641361439410, −17.48199136604139344794241435527, −16.598343634782327680697495685133, −15.877495056886645741814417157, −14.212362986493840608243109079757, −13.54220049762854698703290636987, −12.998096033032114173498715630010, −11.43148721372772450120742544006, −10.54476347146024958740754574667, −9.78893184878775269241091012569, −8.919361499794702341802001895056, −8.26124392346207306647067816410, −6.78519799291964621860142416575, −4.923412477965179843783290705857, −4.263331266446305265553294368320, −3.09393020628284820422516556974, −1.946935936426715327025110051, −0.654139137798256429965466710780,
1.35922508658359381460982393461, 2.414679596168811102713912926047, 3.82990443058791371734724297553, 5.87241483366613851380249065319, 6.045868653678749082499351646388, 7.28191526772871079260406149910, 8.27201275945763544900108845351, 9.09420224902334375977687919145, 9.91554842797905486422329365763, 11.22108010801244105187275658997, 12.75692071499204596644411305157, 13.50291973977043460118934913308, 14.369596716114554056604299456573, 15.138216772841547046752761057463, 15.91986875140583555952613797743, 17.42399356830472310182514401260, 18.01336802072365946571070764042, 18.73500764064785430101351027375, 19.35251380352091276736316680682, 20.670212849889658262154968472227, 21.77437486917443275018874561952, 22.71962207512354801027786005286, 23.63736754912004790533048972178, 24.7202006009805827197450683066, 25.16735237788603669782031240654