L(s) = 1 | + (−0.186 + 0.982i)2-s + (−2.06 − 0.0483i)3-s + (−0.930 − 0.366i)4-s + (−0.765 − 2.90i)5-s + (0.431 − 2.01i)6-s + (0.890 − 1.98i)7-s + (0.533 − 0.845i)8-s + (1.24 + 0.0584i)9-s + (2.99 − 0.210i)10-s + (−0.647 + 0.727i)11-s + (1.89 + 0.799i)12-s + (0.223 + 3.17i)13-s + (1.78 + 1.24i)14-s + (1.43 + 6.01i)15-s + (0.731 + 0.681i)16-s + (−4.38 + 3.89i)17-s + ⋯ |
L(s) = 1 | + (−0.131 + 0.694i)2-s + (−1.18 − 0.0278i)3-s + (−0.465 − 0.183i)4-s + (−0.342 − 1.29i)5-s + (0.176 − 0.822i)6-s + (0.336 − 0.749i)7-s + (0.188 − 0.299i)8-s + (0.415 + 0.0194i)9-s + (0.947 − 0.0667i)10-s + (−0.195 + 0.219i)11-s + (0.548 + 0.230i)12-s + (0.0620 + 0.881i)13-s + (0.476 + 0.332i)14-s + (0.371 + 1.55i)15-s + (0.182 + 0.170i)16-s + (−1.06 + 0.945i)17-s + ⋯ |
Λ(s)=(=(538s/2ΓC(s)L(s)(−0.934−0.356i)Λ(2−s)
Λ(s)=(=(538s/2ΓC(s+1/2)L(s)(−0.934−0.356i)Λ(1−s)
Degree: |
2 |
Conductor: |
538
= 2⋅269
|
Sign: |
−0.934−0.356i
|
Analytic conductor: |
4.29595 |
Root analytic conductor: |
2.07266 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ538(369,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 538, ( :1/2), −0.934−0.356i)
|
Particular Values
L(1) |
≈ |
0.0249287+0.135341i |
L(21) |
≈ |
0.0249287+0.135341i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.186−0.982i)T |
| 269 | 1+(15.4+5.60i)T |
good | 3 | 1+(2.06+0.0483i)T+(2.99+0.140i)T2 |
| 5 | 1+(0.765+2.90i)T+(−4.34+2.46i)T2 |
| 7 | 1+(−0.890+1.98i)T+(−4.65−5.23i)T2 |
| 11 | 1+(0.647−0.727i)T+(−1.28−10.9i)T2 |
| 13 | 1+(−0.223−3.17i)T+(−12.8+1.82i)T2 |
| 17 | 1+(4.38−3.89i)T+(1.98−16.8i)T2 |
| 19 | 1+(1.29−0.857i)T+(7.37−17.5i)T2 |
| 23 | 1+(−0.0751+3.20i)T+(−22.9−1.07i)T2 |
| 29 | 1+(2.61−4.60i)T+(−14.8−24.8i)T2 |
| 31 | 1+(2.84−0.335i)T+(30.1−7.20i)T2 |
| 37 | 1+(1.28+2.39i)T+(−20.4+30.8i)T2 |
| 41 | 1+(2.32−0.440i)T+(38.1−15.0i)T2 |
| 43 | 1+(−4.44−2.52i)T+(22.0+36.8i)T2 |
| 47 | 1+(0.272−0.0788i)T+(39.7−25.0i)T2 |
| 53 | 1+(7.98−10.8i)T+(−15.9−50.5i)T2 |
| 59 | 1+(1.59−4.05i)T+(−43.1−40.2i)T2 |
| 61 | 1+(6.91−8.54i)T+(−12.7−59.6i)T2 |
| 67 | 1+(−2.68+1.05i)T+(49.0−45.6i)T2 |
| 71 | 1+(6.75+9.67i)T+(−24.4+66.6i)T2 |
| 73 | 1+(−0.149−0.406i)T+(−55.6+47.2i)T2 |
| 79 | 1+(1.94−1.50i)T+(20.1−76.3i)T2 |
| 83 | 1+(0.804−8.54i)T+(−81.5−15.4i)T2 |
| 89 | 1+(−0.744+4.49i)T+(−84.2−28.6i)T2 |
| 97 | 1+(−6.95−8.60i)T+(−20.3+94.8i)T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.04268486670315221085120683192, −10.58467207024498496898738264254, −9.172762296013599672881691532517, −8.581080355594606761400401526056, −7.52988963828064339581071034240, −6.58101763522647673293673202024, −5.70296782862142400267808186694, −4.63285235947857113174186276613, −4.25804613242130860304309800052, −1.39240794649207275132172270612,
0.10192917759816884697659065899, 2.34113347458488596798725849391, 3.36978791787020546121287676388, 4.84695977152715975099902385786, 5.71607233314443666787823721410, 6.67272522681332599868755867190, 7.68866392861177054013052489512, 8.780894744811539387548355578483, 9.911700947055786357085789839542, 10.79766353062664221048410368796