L(s) = 1 | + (−0.885 − 0.464i)3-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.568 + 0.822i)9-s + (0.822 + 0.568i)11-s + (0.239 + 0.970i)15-s + (0.970 − 0.239i)17-s − i·19-s + (−0.239 + 0.970i)21-s + 23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.568 + 0.822i)29-s + (0.992 − 0.120i)31-s + (−0.464 − 0.885i)33-s + ⋯ |
L(s) = 1 | + (−0.885 − 0.464i)3-s + (−0.663 − 0.748i)5-s + (−0.239 − 0.970i)7-s + (0.568 + 0.822i)9-s + (0.822 + 0.568i)11-s + (0.239 + 0.970i)15-s + (0.970 − 0.239i)17-s − i·19-s + (−0.239 + 0.970i)21-s + 23-s + (−0.120 + 0.992i)25-s + (−0.120 − 0.992i)27-s + (0.568 + 0.822i)29-s + (0.992 − 0.120i)31-s + (−0.464 − 0.885i)33-s + ⋯ |
Λ(s)=(=(676s/2ΓR(s)L(s)(−0.175−0.984i)Λ(1−s)
Λ(s)=(=(676s/2ΓR(s)L(s)(−0.175−0.984i)Λ(1−s)
Degree: |
1 |
Conductor: |
676
= 22⋅132
|
Sign: |
−0.175−0.984i
|
Analytic conductor: |
3.13933 |
Root analytic conductor: |
3.13933 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ676(47,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 676, (0: ), −0.175−0.984i)
|
Particular Values
L(21) |
≈ |
0.6003362776−0.7169553146i |
L(21) |
≈ |
0.6003362776−0.7169553146i |
L(1) |
≈ |
0.7315017283−0.3223538015i |
L(1) |
≈ |
0.7315017283−0.3223538015i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1 |
good | 3 | 1+(−0.885−0.464i)T |
| 5 | 1+(−0.663−0.748i)T |
| 7 | 1+(−0.239−0.970i)T |
| 11 | 1+(0.822+0.568i)T |
| 17 | 1+(0.970−0.239i)T |
| 19 | 1−iT |
| 23 | 1+T |
| 29 | 1+(0.568+0.822i)T |
| 31 | 1+(0.992−0.120i)T |
| 37 | 1+(0.992−0.120i)T |
| 41 | 1+(−0.464+0.885i)T |
| 43 | 1+(0.120−0.992i)T |
| 47 | 1+(0.935−0.354i)T |
| 53 | 1+(−0.970+0.239i)T |
| 59 | 1+(−0.663−0.748i)T |
| 61 | 1+(−0.970−0.239i)T |
| 67 | 1+(−0.935+0.354i)T |
| 71 | 1+(0.464−0.885i)T |
| 73 | 1+(−0.822−0.568i)T |
| 79 | 1+(0.354+0.935i)T |
| 83 | 1+(−0.464−0.885i)T |
| 89 | 1−iT |
| 97 | 1+(0.663−0.748i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−22.89067091228411552516001962554, −22.25158057031116773056695991002, −21.51922864055352068663284799702, −20.816727008888775907448069876171, −19.3510257369414957068745343190, −18.93025263000201176238533534357, −18.15519787112302205167726655619, −17.09478222483286349446421813518, −16.39502717898574402153510755713, −15.53644366696405221351536912602, −14.925890381359816333241555516996, −14.08080551331659637010929548905, −12.58031799543151122989725058063, −11.94180833012883884423814431976, −11.37267386812766901347771107948, −10.419327934620607597661985602381, −9.612199669900761504016109355563, −8.57259567315531574274417471051, −7.50741674128456452812125255915, −6.27136204611452497478621764728, −5.97106779166208142084108631110, −4.67311354288821745639293260850, −3.665623282156886884532442277953, −2.85370802692362787602399456012, −1.116257865844979678858322427452,
0.68610962558817533855120446913, 1.418828151152139105436081783889, 3.20418582751835058582267865769, 4.46465510554987139557577397156, 4.89463794164094366507002406926, 6.244103366292967015106434175128, 7.13706640450052574030294240893, 7.688431746905325337670384847395, 8.93911891251075185328287001983, 9.9342688946259410217557586733, 10.918351278450922548549936489769, 11.71881576960149565945341624320, 12.430194058888386441029695196426, 13.16014103481084625199680541080, 14.04401858470561226607716412295, 15.26384035405670190940315898505, 16.15670273040683097517182665061, 17.01040631590041854989270691501, 17.20502713200000745273391808555, 18.42055167758261468102446747427, 19.38674548865680747183403431124, 19.91392786339231233054087919378, 20.79374596431632758662307097970, 21.8690691328774537958997787791, 22.789988567252438705594645383476