L(s) = 1 | − 2.65i·3-s − 5-s − 1.97i·7-s − 4.04·9-s + 5.81i·11-s + 1.97i·13-s + 2.65i·15-s − 2.65i·17-s + 8.26i·19-s − 5.23·21-s − 3.04·23-s + 25-s + 2.76i·27-s + 7.39i·29-s − 6.49·31-s + ⋯ |
L(s) = 1 | − 1.53i·3-s − 0.447·5-s − 0.745i·7-s − 1.34·9-s + 1.75i·11-s + 0.546i·13-s + 0.685i·15-s − 0.643i·17-s + 1.89i·19-s − 1.14·21-s − 0.634·23-s + 0.200·25-s + 0.532i·27-s + 1.37i·29-s − 1.16·31-s + ⋯ |
Λ(s)=(=(1640s/2ΓC(s)L(s)(0.672−0.740i)Λ(2−s)
Λ(s)=(=(1640s/2ΓC(s+1/2)L(s)(0.672−0.740i)Λ(1−s)
Degree: |
2 |
Conductor: |
1640
= 23⋅5⋅41
|
Sign: |
0.672−0.740i
|
Analytic conductor: |
13.0954 |
Root analytic conductor: |
3.61876 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1640(81,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1640, ( :1/2), 0.672−0.740i)
|
Particular Values
L(1) |
≈ |
0.7682168435 |
L(21) |
≈ |
0.7682168435 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+T |
| 41 | 1+(4.30−4.74i)T |
good | 3 | 1+2.65iT−3T2 |
| 7 | 1+1.97iT−7T2 |
| 11 | 1−5.81iT−11T2 |
| 13 | 1−1.97iT−13T2 |
| 17 | 1+2.65iT−17T2 |
| 19 | 1−8.26iT−19T2 |
| 23 | 1+3.04T+23T2 |
| 29 | 1−7.39iT−29T2 |
| 31 | 1+6.49T+31T2 |
| 37 | 1−1.45T+37T2 |
| 43 | 1+11.3T+43T2 |
| 47 | 1−10.0iT−47T2 |
| 53 | 1+9.33iT−53T2 |
| 59 | 1−7.15T+59T2 |
| 61 | 1−9.72T+61T2 |
| 67 | 1+1.15iT−67T2 |
| 71 | 1+4.05iT−71T2 |
| 73 | 1+1.59T+73T2 |
| 79 | 1+1.57iT−79T2 |
| 83 | 1−4.11T+83T2 |
| 89 | 1−17.9iT−89T2 |
| 97 | 1+12.8iT−97T2 |
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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
−9.558536054423451599104148525407, −8.290933198029373123355228148285, −7.78083026645325736212266234364, −6.93575743057629344395532924264, −6.82357110662940211354406547995, −5.51015308869037168052884429165, −4.45225240480219054903571784719, −3.49983361075254148535155330038, −2.03546466560503920474742035875, −1.40375150233782228300218548901,
0.29619079096306906098490376080, 2.56015149172486944881949137532, 3.46399976711725860526999046659, 4.09745345521367322470836044310, 5.26000144813132876462206314397, 5.64403730222312693147208662376, 6.72959813393347939876287851724, 8.071793308220773874806007203822, 8.687358924920554430164754236628, 9.139971828492265564348647188568