L(s) = 1 | + (0.998 + 0.0468i)2-s + (0.388 − 0.921i)3-s + (0.995 + 0.0936i)4-s + (−0.946 − 0.322i)5-s + (0.430 − 0.902i)6-s + (0.628 + 0.777i)7-s + (0.990 + 0.140i)8-s + (−0.698 − 0.715i)9-s + (−0.930 − 0.366i)10-s + (−0.209 − 0.977i)11-s + (0.472 − 0.881i)12-s + (0.930 + 0.366i)13-s + (0.591 + 0.806i)14-s + (−0.664 + 0.747i)15-s + (0.982 + 0.186i)16-s + (0.209 − 0.977i)17-s + ⋯ |
L(s) = 1 | + (0.998 + 0.0468i)2-s + (0.388 − 0.921i)3-s + (0.995 + 0.0936i)4-s + (−0.946 − 0.322i)5-s + (0.430 − 0.902i)6-s + (0.628 + 0.777i)7-s + (0.990 + 0.140i)8-s + (−0.698 − 0.715i)9-s + (−0.930 − 0.366i)10-s + (−0.209 − 0.977i)11-s + (0.472 − 0.881i)12-s + (0.930 + 0.366i)13-s + (0.591 + 0.806i)14-s + (−0.664 + 0.747i)15-s + (0.982 + 0.186i)16-s + (0.209 − 0.977i)17-s + ⋯ |
Λ(s)=(=(269s/2ΓR(s)L(s)(0.550−0.834i)Λ(1−s)
Λ(s)=(=(269s/2ΓR(s)L(s)(0.550−0.834i)Λ(1−s)
Degree: |
1 |
Conductor: |
269
|
Sign: |
0.550−0.834i
|
Analytic conductor: |
1.24923 |
Root analytic conductor: |
1.24923 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ269(4,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(1, 269, (0: ), 0.550−0.834i)
|
Particular Values
L(21) |
≈ |
2.076179029−1.117450581i |
L(21) |
≈ |
2.076179029−1.117450581i |
L(1) |
≈ |
1.840258165−0.5841070263i |
L(1) |
≈ |
1.840258165−0.5841070263i |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 269 | 1 |
good | 2 | 1+(0.998+0.0468i)T |
| 3 | 1+(0.388−0.921i)T |
| 5 | 1+(−0.946−0.322i)T |
| 7 | 1+(0.628+0.777i)T |
| 11 | 1+(−0.209−0.977i)T |
| 13 | 1+(0.930+0.366i)T |
| 17 | 1+(0.209−0.977i)T |
| 19 | 1+(−0.513−0.858i)T |
| 23 | 1+(−0.388+0.921i)T |
| 29 | 1+(−0.792+0.610i)T |
| 31 | 1+(0.912−0.409i)T |
| 37 | 1+(−0.869+0.493i)T |
| 41 | 1+(−0.998+0.0468i)T |
| 43 | 1+(0.792−0.610i)T |
| 47 | 1+(0.0702+0.997i)T |
| 53 | 1+(−0.972+0.232i)T |
| 59 | 1+(−0.995−0.0936i)T |
| 61 | 1+(0.845+0.533i)T |
| 67 | 1+(0.995−0.0936i)T |
| 71 | 1+(−0.591+0.806i)T |
| 73 | 1+(−0.300+0.953i)T |
| 79 | 1+(0.163+0.986i)T |
| 83 | 1+(0.0234−0.999i)T |
| 89 | 1+(0.344+0.938i)T |
| 97 | 1+(0.845−0.533i)T |
show more | |
show less | |
L(s)=p∏ (1−αpp−s)−1
Imaginary part of the first few zeros on the critical line
−25.99577843301047012561067726947, −24.960368174414643090144439138233, −23.74954075755303160634010698613, −23.04388854876679469394453648480, −22.500349324440626789300116654607, −21.17185707973282832674307173999, −20.604055069017795035755565317813, −19.962076738619059106841746418076, −18.91846011930386661958962737119, −17.28177530051317734705599687694, −16.29538737264517101752719428228, −15.40521590303126708388284194088, −14.79145194174743928563880190564, −14.04632395378409744696113492657, −12.821202463742710962695017174005, −11.73582852904819455156924289632, −10.62716612992833987896751907390, −10.34345978686311443253549991746, −8.31092534054113962049971473470, −7.68175879789422856722149010818, −6.31964716235086825004658959134, −4.893045461791652230202041358064, −4.06757295799290249517492728977, −3.478891796542804760213079816144, −1.945129200592844655981097469585,
1.330975446850984347228111706325, 2.708306841392677222814112800669, 3.66579664391080428908789461485, 5.037135052175373985355358742453, 6.057039942543714273356327885731, 7.20721665514897719274689539646, 8.14635834627501838019786903216, 8.905322201704234269201565783295, 11.242809085665393093276201962340, 11.533794360967725662153020610315, 12.516121963436879085287402718554, 13.49902065184676139857969160914, 14.20398672258807065575167691003, 15.376652376370816742408949602675, 15.930235750214468394937733411, 17.24953800295583863220017932229, 18.66345920337516423385568423327, 19.16079620622259493390744049612, 20.339105664413777036133302853882, 20.95326463987160435018595895340, 22.05878465654332331396724844133, 23.20187720204804670101844021489, 23.97658916500742528370446685908, 24.30106121912864773802845596909, 25.33249035029173838903359934505