L(s) = 1 | + (−3.20 − 0.858i)2-s + (1.5 + 2.59i)3-s + (6.06 + 3.5i)4-s + (−2.34 + 2.34i)5-s + (−2.57 − 9.61i)6-s + (9.61 − 2.57i)7-s + (−7.03 − 7.03i)8-s + (9.52 − 5.5i)10-s + (−1.71 + 6.40i)11-s + 21i·12-s − 33·14-s + (−9.61 − 2.57i)15-s + (2.49 + 4.33i)16-s + (2.59 + 1.5i)17-s + (5.15 + 19.2i)19-s + (−22.4 + 6.00i)20-s + ⋯ |
L(s) = 1 | + (−1.60 − 0.429i)2-s + (0.5 + 0.866i)3-s + (1.51 + 0.875i)4-s + (−0.469 + 0.469i)5-s + (−0.429 − 1.60i)6-s + (1.37 − 0.367i)7-s + (−0.879 − 0.879i)8-s + (0.952 − 0.550i)10-s + (−0.156 + 0.582i)11-s + 1.75i·12-s − 2.35·14-s + (−0.640 − 0.171i)15-s + (0.156 + 0.270i)16-s + (0.152 + 0.0882i)17-s + (0.271 + 1.01i)19-s + (−1.12 + 0.300i)20-s + ⋯ |
Λ(s)=(=(169s/2ΓC(s)L(s)(0.466−0.884i)Λ(3−s)
Λ(s)=(=(169s/2ΓC(s+1)L(s)(0.466−0.884i)Λ(1−s)
Degree: |
2 |
Conductor: |
169
= 132
|
Sign: |
0.466−0.884i
|
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), 0.466−0.884i)
|
Particular Values
L(23) |
≈ |
0.718296+0.433410i |
L(21) |
≈ |
0.718296+0.433410i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1 |
good | 2 | 1+(3.20+0.858i)T+(3.46+2i)T2 |
| 3 | 1+(−1.5−2.59i)T+(−4.5+7.79i)T2 |
| 5 | 1+(2.34−2.34i)T−25iT2 |
| 7 | 1+(−9.61+2.57i)T+(42.4−24.5i)T2 |
| 11 | 1+(1.71−6.40i)T+(−104.−60.5i)T2 |
| 17 | 1+(−2.59−1.5i)T+(144.5+250.i)T2 |
| 19 | 1+(−5.15−19.2i)T+(−312.+180.5i)T2 |
| 23 | 1+(10.3−6i)T+(264.5−458.i)T2 |
| 29 | 1+(−21−36.3i)T+(−420.5+728.i)T2 |
| 31 | 1+(−28.1+28.1i)T−961iT2 |
| 37 | 1+(12.8−48.0i)T+(−1.18e3−684.5i)T2 |
| 41 | 1+(44.8+12.0i)T+(1.45e3+840.5i)T2 |
| 43 | 1+(42.4+24.5i)T+(924.5+1.60e3i)T2 |
| 47 | 1+(−2.34−2.34i)T+2.20e3iT2 |
| 53 | 1+24T+2.80e3T2 |
| 59 | 1+(−51.2+13.7i)T+(3.01e3−1.74e3i)T2 |
| 61 | 1+(15−25.9i)T+(−1.86e3−3.22e3i)T2 |
| 67 | 1+(−38.4−10.3i)T+(3.88e3+2.24e3i)T2 |
| 71 | 1+(6.00+22.4i)T+(−4.36e3+2.52e3i)T2 |
| 73 | 1+(28.1+28.1i)T+5.32e3iT2 |
| 79 | 1−54T+6.24e3T2 |
| 83 | 1+(32.8−32.8i)T−6.88e3iT2 |
| 89 | 1+(−42.9+160.i)T+(−6.85e3−3.96e3i)T2 |
| 97 | 1+(−5.15−19.2i)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
−12.08809313583924104626077259694, −11.36549564586994729918974358352, −10.32640702401923004293301019143, −9.933786352359915190939843187195, −8.636798287074699358204306753318, −7.989811449608565091429681007804, −7.03998232128868701594060047431, −4.76571077294677210905156305182, −3.35414497872606038237715613860, −1.59271636956302383774188862051,
0.912020867882832893194658233092, 2.24640555134595149452501706415, 4.85643807956893440104572000467, 6.55557100448021722005814690858, 7.70238840865830561390739262683, 8.264698902656337500075462757401, 8.752026978253210029948766202367, 10.18366524353680606810043995815, 11.27057705638109144170511252943, 12.08158537243959159751942888048