L(s) = 1 | + (1.92 + 0.515i)2-s + (0.299 + 0.519i)3-s + (−0.0309 − 0.0178i)4-s + (5.65 − 5.65i)5-s + (0.308 + 1.15i)6-s + (1.96 − 0.526i)7-s + (−5.68 − 5.68i)8-s + (4.32 − 7.48i)9-s + (13.7 − 7.95i)10-s + (−3.76 + 14.0i)11-s − 0.0214i·12-s + 4.05·14-s + (4.62 + 1.24i)15-s + (−7.92 − 13.7i)16-s + (10.2 + 5.90i)17-s + (12.1 − 12.1i)18-s + ⋯ |
L(s) = 1 | + (0.961 + 0.257i)2-s + (0.0999 + 0.173i)3-s + (−0.00773 − 0.00446i)4-s + (1.13 − 1.13i)5-s + (0.0514 + 0.192i)6-s + (0.280 − 0.0752i)7-s + (−0.710 − 0.710i)8-s + (0.480 − 0.831i)9-s + (1.37 − 0.795i)10-s + (−0.342 + 1.27i)11-s − 0.00178i·12-s + 0.289·14-s + (0.308 + 0.0826i)15-s + (−0.495 − 0.858i)16-s + (0.601 + 0.347i)17-s + (0.675 − 0.675i)18-s + ⋯ |
Λ(s)=(=(169s/2ΓC(s)L(s)(0.940+0.338i)Λ(3−s)
Λ(s)=(=(169s/2ΓC(s+1)L(s)(0.940+0.338i)Λ(1−s)
Degree: |
2 |
Conductor: |
169
= 132
|
Sign: |
0.940+0.338i
|
Analytic conductor: |
4.60491 |
Root analytic conductor: |
2.14590 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ169(80,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 169, ( :1), 0.940+0.338i)
|
Particular Values
L(23) |
≈ |
2.57907−0.450333i |
L(21) |
≈ |
2.57907−0.450333i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1 |
good | 2 | 1+(−1.92−0.515i)T+(3.46+2i)T2 |
| 3 | 1+(−0.299−0.519i)T+(−4.5+7.79i)T2 |
| 5 | 1+(−5.65+5.65i)T−25iT2 |
| 7 | 1+(−1.96+0.526i)T+(42.4−24.5i)T2 |
| 11 | 1+(3.76−14.0i)T+(−104.−60.5i)T2 |
| 17 | 1+(−10.2−5.90i)T+(144.5+250.i)T2 |
| 19 | 1+(−6.28−23.4i)T+(−312.+180.5i)T2 |
| 23 | 1+(0.229−0.132i)T+(264.5−458.i)T2 |
| 29 | 1+(3.60+6.25i)T+(−420.5+728.i)T2 |
| 31 | 1+(30.2−30.2i)T−961iT2 |
| 37 | 1+(3.53−13.1i)T+(−1.18e3−684.5i)T2 |
| 41 | 1+(33.6+9.00i)T+(1.45e3+840.5i)T2 |
| 43 | 1+(35.0+20.2i)T+(924.5+1.60e3i)T2 |
| 47 | 1+(9.87+9.87i)T+2.20e3iT2 |
| 53 | 1−77.1T+2.80e3T2 |
| 59 | 1+(−34.8+9.32i)T+(3.01e3−1.74e3i)T2 |
| 61 | 1+(−15.4+26.7i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(−60.4−16.1i)T+(3.88e3+2.24e3i)T2 |
| 71 | 1+(2.23+8.34i)T+(−4.36e3+2.52e3i)T2 |
| 73 | 1+(−23.3−23.3i)T+5.32e3iT2 |
| 79 | 1+49.8T+6.24e3T2 |
| 83 | 1+(60.7−60.7i)T−6.88e3iT2 |
| 89 | 1+(22.9−85.6i)T+(−6.85e3−3.96e3i)T2 |
| 97 | 1+(−8.75−32.6i)T+(−8.14e3+4.70e3i)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
−12.56830287034276709006325616048, −12.21609342870950138266742342185, −10.00774344372366703959544937752, −9.722247914219591918664422937848, −8.541554252623482829451525207021, −6.90808866962831181460650825295, −5.64957993331227634182105829143, −4.94767628470369084357770893369, −3.79599397496935433433917736045, −1.53555852341215507486665569055,
2.30516251637738829845594621606, 3.30604739676516240317823356619, 5.06250210441236490017404549784, 5.87576735630183454618574877818, 7.16811059324068860971981135498, 8.492086121034477012279525946456, 9.772594621487086659906018557689, 10.88270450775610632024779267822, 11.53686150661793959961383003711, 13.15039869545051386122149859649