L(s) = 1 | + (−1.18 + 0.864i)2-s + (−0.639 − 1.96i)3-s + (0.0502 − 0.154i)4-s + (0.989 + 0.718i)5-s + (2.46 + 1.78i)6-s + (−0.728 + 2.24i)7-s + (−0.834 − 2.56i)8-s + (−1.03 + 0.753i)9-s − 1.79·10-s + (3.28 + 0.479i)11-s − 0.336·12-s + (2.06 − 1.50i)13-s + (−1.07 − 3.29i)14-s + (0.782 − 2.40i)15-s + (3.47 + 2.52i)16-s + (5.81 + 4.22i)17-s + ⋯ |
L(s) = 1 | + (−0.841 + 0.611i)2-s + (−0.369 − 1.13i)3-s + (0.0251 − 0.0773i)4-s + (0.442 + 0.321i)5-s + (1.00 + 0.730i)6-s + (−0.275 + 0.847i)7-s + (−0.295 − 0.908i)8-s + (−0.345 + 0.251i)9-s − 0.568·10-s + (0.989 + 0.144i)11-s − 0.0971·12-s + (0.573 − 0.416i)13-s + (−0.286 − 0.881i)14-s + (0.201 − 0.621i)15-s + (0.869 + 0.631i)16-s + (1.40 + 1.02i)17-s + ⋯ |
Λ(s)=(=(209s/2ΓC(s)L(s)(0.896−0.443i)Λ(2−s)
Λ(s)=(=(209s/2ΓC(s+1/2)L(s)(0.896−0.443i)Λ(1−s)
Degree: |
2 |
Conductor: |
209
= 11⋅19
|
Sign: |
0.896−0.443i
|
Analytic conductor: |
1.66887 |
Root analytic conductor: |
1.29184 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ209(58,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 209, ( :1/2), 0.896−0.443i)
|
Particular Values
L(1) |
≈ |
0.749569+0.175321i |
L(21) |
≈ |
0.749569+0.175321i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 11 | 1+(−3.28−0.479i)T |
| 19 | 1+(−0.309−0.951i)T |
good | 2 | 1+(1.18−0.864i)T+(0.618−1.90i)T2 |
| 3 | 1+(0.639+1.96i)T+(−2.42+1.76i)T2 |
| 5 | 1+(−0.989−0.718i)T+(1.54+4.75i)T2 |
| 7 | 1+(0.728−2.24i)T+(−5.66−4.11i)T2 |
| 13 | 1+(−2.06+1.50i)T+(4.01−12.3i)T2 |
| 17 | 1+(−5.81−4.22i)T+(5.25+16.1i)T2 |
| 23 | 1−6.34T+23T2 |
| 29 | 1+(1.34−4.13i)T+(−23.4−17.0i)T2 |
| 31 | 1+(3.49−2.53i)T+(9.57−29.4i)T2 |
| 37 | 1+(−2.25+6.93i)T+(−29.9−21.7i)T2 |
| 41 | 1+(−3.36−10.3i)T+(−33.1+24.0i)T2 |
| 43 | 1+7.28T+43T2 |
| 47 | 1+(3.66+11.2i)T+(−38.0+27.6i)T2 |
| 53 | 1+(0.504−0.366i)T+(16.3−50.4i)T2 |
| 59 | 1+(−0.0443+0.136i)T+(−47.7−34.6i)T2 |
| 61 | 1+(4.33+3.15i)T+(18.8+58.0i)T2 |
| 67 | 1+4.80T+67T2 |
| 71 | 1+(2.65+1.93i)T+(21.9+67.5i)T2 |
| 73 | 1+(0.682−2.09i)T+(−59.0−42.9i)T2 |
| 79 | 1+(6.95−5.05i)T+(24.4−75.1i)T2 |
| 83 | 1+(4.83+3.51i)T+(25.6+78.9i)T2 |
| 89 | 1−10.0T+89T2 |
| 97 | 1+(−1.12+0.814i)T+(29.9−92.2i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−12.59471000201501407332134697401, −11.69131270631688254814310135142, −10.26536913102522906447413955310, −9.257108602961692844448645886600, −8.361562022966260147223592632114, −7.34291258980162346704368768257, −6.44261365935631967519894617458, −5.80363309874037569977914708919, −3.41968186120199046747053025971, −1.38694841499191491090714084908,
1.19486150149946947123931596337, 3.44928968536212069288417111523, 4.77867910442720727523813922218, 5.84663296464850856499260815033, 7.40577270716483326369156306120, 9.017275837066384870314198854774, 9.510060487535669508431034425510, 10.19872300734712040257834668017, 11.11886071658120121042360608876, 11.73245205186603294801165260099