L(s) = 1 | + (−0.581 + 0.813i)2-s + (−0.976 − 0.217i)3-s + (−0.322 − 0.946i)4-s + (0.639 − 0.768i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (0.957 + 0.288i)8-s + (0.905 + 0.424i)9-s + (0.252 + 0.967i)10-s + (0.520 + 0.853i)11-s + (0.109 + 0.994i)12-s + (0.581 + 0.813i)13-s + (0.872 − 0.489i)14-s + (−0.791 + 0.611i)15-s + (−0.791 + 0.611i)16-s + (−0.520 − 0.853i)17-s + ⋯ |
L(s) = 1 | + (−0.581 + 0.813i)2-s + (−0.976 − 0.217i)3-s + (−0.322 − 0.946i)4-s + (0.639 − 0.768i)5-s + (0.744 − 0.667i)6-s + (−0.905 − 0.424i)7-s + (0.957 + 0.288i)8-s + (0.905 + 0.424i)9-s + (0.252 + 0.967i)10-s + (0.520 + 0.853i)11-s + (0.109 + 0.994i)12-s + (0.581 + 0.813i)13-s + (0.872 − 0.489i)14-s + (−0.791 + 0.611i)15-s + (−0.791 + 0.611i)16-s + (−0.520 − 0.853i)17-s + ⋯ |
Λ(s)=(=(431s/2ΓR(s+1)L(s)(−0.114+0.993i)Λ(1−s)
Λ(s)=(=(431s/2ΓR(s+1)L(s)(−0.114+0.993i)Λ(1−s)
Degree: |
1 |
Conductor: |
431
|
Sign: |
−0.114+0.993i
|
Analytic conductor: |
46.3173 |
Root analytic conductor: |
46.3173 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ431(367,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 431, (1: ), −0.114+0.993i)
|
Particular Values
L(21) |
≈ |
0.4531352824+0.5084804492i |
L(21) |
≈ |
0.4531352824+0.5084804492i |
L(1) |
≈ |
0.5796830088+0.1267796929i |
L(1) |
≈ |
0.5796830088+0.1267796929i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 431 | 1 |
good | 2 | 1+(−0.581+0.813i)T |
| 3 | 1+(−0.976−0.217i)T |
| 5 | 1+(0.639−0.768i)T |
| 7 | 1+(−0.905−0.424i)T |
| 11 | 1+(0.520+0.853i)T |
| 13 | 1+(0.581+0.813i)T |
| 17 | 1+(−0.520−0.853i)T |
| 19 | 1+(0.639+0.768i)T |
| 23 | 1+(−0.0365−0.999i)T |
| 29 | 1+(−0.322+0.946i)T |
| 31 | 1+(−0.744−0.667i)T |
| 37 | 1+(0.872−0.489i)T |
| 41 | 1+(−0.791+0.611i)T |
| 43 | 1+(−0.252+0.967i)T |
| 47 | 1+(−0.989+0.145i)T |
| 53 | 1+(0.989+0.145i)T |
| 59 | 1+(−0.581+0.813i)T |
| 61 | 1+(−0.694−0.719i)T |
| 67 | 1+(0.872−0.489i)T |
| 71 | 1+(−0.989+0.145i)T |
| 73 | 1+(−0.391−0.920i)T |
| 79 | 1+(−0.833+0.551i)T |
| 83 | 1+(0.976+0.217i)T |
| 89 | 1+(0.934+0.357i)T |
| 97 | 1+(0.833+0.551i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−23.3219859774471935958058063443, −22.444486646877790610444131691012, −21.87641327483486464704154181685, −21.50652357483713015035318990433, −20.16007182230975209542186314915, −19.18081085287920758966878416644, −18.48000717244407418499972206988, −17.70573826557487865038619681566, −17.04608342985136205716872738287, −16.04395028241143748280506970767, −15.20921810132060322001630035248, −13.49665600180046084382113792881, −13.08499426974492885210740640572, −11.83122375514205885786744758040, −11.151153975119400456303733691232, −10.35813748852968084070271617467, −9.61813151664264416437334221562, −8.72266240450363159276766920189, −7.20832261525262466745142597609, −6.266404484287868325411459578241, −5.46680958431954477206429614341, −3.771426158136544956701515688046, −3.07798442829716368398996184886, −1.641436984638840286186835872540, −0.33450073360257433565298754872,
0.89879251394723425784393922605, 1.83596406010170896389605595472, 4.20809433026769274678871054423, 5.02335106616191417405280780990, 6.1595546855002009544894627062, 6.65966683070242713274637630487, 7.629406883218984912095104814778, 9.120588293400596742799264135307, 9.62874465801180879165644085793, 10.52929702432839516911906102759, 11.71966852318756127447844878332, 12.83597781529211140018259746536, 13.52480922289832581182265317325, 14.57467869010000778832689979411, 16.00332772576868039653040347519, 16.49860175195297351336064908283, 16.926686913782899760482604738962, 18.11758535312219673699477645946, 18.43817133550442010821400032521, 19.79339577580632745209913225705, 20.484908254777856498600620158953, 21.89430597944708043069211313198, 22.75035439374288407205342680275, 23.29766607757294342778765622896, 24.32088034503486797845544290654