L(s) = 1 | + (1.61 − 2.02i)2-s + (−0.222 + 0.974i)3-s + (−1.04 − 4.58i)4-s + (−0.716 + 0.898i)5-s + (1.61 + 2.02i)6-s + (−0.615 + 2.69i)7-s + (−6.29 − 3.03i)8-s + (−0.900 − 0.433i)9-s + (0.661 + 2.89i)10-s + (3.92 − 1.88i)11-s + 4.70·12-s + (−5.07 + 2.44i)13-s + (4.46 + 5.59i)14-s + (−0.716 − 0.898i)15-s + (−7.83 + 3.77i)16-s + 2.41·17-s + ⋯ |
L(s) = 1 | + (1.14 − 1.43i)2-s + (−0.128 + 0.562i)3-s + (−0.523 − 2.29i)4-s + (−0.320 + 0.401i)5-s + (0.658 + 0.826i)6-s + (−0.232 + 1.01i)7-s + (−2.22 − 1.07i)8-s + (−0.300 − 0.144i)9-s + (0.209 + 0.916i)10-s + (1.18 − 0.569i)11-s + 1.35·12-s + (−1.40 + 0.677i)13-s + (1.19 + 1.49i)14-s + (−0.184 − 0.231i)15-s + (−1.95 + 0.943i)16-s + 0.585·17-s + ⋯ |
Λ(s)=(=(87s/2ΓC(s)L(s)(0.238+0.971i)Λ(2−s)
Λ(s)=(=(87s/2ΓC(s+1/2)L(s)(0.238+0.971i)Λ(1−s)
Degree: |
2 |
Conductor: |
87
= 3⋅29
|
Sign: |
0.238+0.971i
|
Analytic conductor: |
0.694698 |
Root analytic conductor: |
0.833485 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ87(52,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 87, ( :1/2), 0.238+0.971i)
|
Particular Values
L(1) |
≈ |
1.12695−0.883566i |
L(21) |
≈ |
1.12695−0.883566i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(0.222−0.974i)T |
| 29 | 1+(−4.53−2.90i)T |
good | 2 | 1+(−1.61+2.02i)T+(−0.445−1.94i)T2 |
| 5 | 1+(0.716−0.898i)T+(−1.11−4.87i)T2 |
| 7 | 1+(0.615−2.69i)T+(−6.30−3.03i)T2 |
| 11 | 1+(−3.92+1.88i)T+(6.85−8.60i)T2 |
| 13 | 1+(5.07−2.44i)T+(8.10−10.1i)T2 |
| 17 | 1−2.41T+17T2 |
| 19 | 1+(0.416+1.82i)T+(−17.1+8.24i)T2 |
| 23 | 1+(5.49+6.89i)T+(−5.11+22.4i)T2 |
| 31 | 1+(−0.972+1.22i)T+(−6.89−30.2i)T2 |
| 37 | 1+(6.99+3.36i)T+(23.0+28.9i)T2 |
| 41 | 1−3.16T+41T2 |
| 43 | 1+(0.912+1.14i)T+(−9.56+41.9i)T2 |
| 47 | 1+(0.321−0.155i)T+(29.3−36.7i)T2 |
| 53 | 1+(1.01−1.27i)T+(−11.7−51.6i)T2 |
| 59 | 1+4.85T+59T2 |
| 61 | 1+(0.346−1.51i)T+(−54.9−26.4i)T2 |
| 67 | 1+(−8.71−4.19i)T+(41.7+52.3i)T2 |
| 71 | 1+(−7.37+3.54i)T+(44.2−55.5i)T2 |
| 73 | 1+(−7.09−8.89i)T+(−16.2+71.1i)T2 |
| 79 | 1+(−7.32−3.52i)T+(49.2+61.7i)T2 |
| 83 | 1+(−0.107−0.469i)T+(−74.7+36.0i)T2 |
| 89 | 1+(1.24−1.56i)T+(−19.8−86.7i)T2 |
| 97 | 1+(1.03+4.51i)T+(−87.3+42.0i)T2 |
show more | |
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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
−14.17287562518573824608123885542, −12.37568178351223711068871158640, −12.02828332861137323908720052767, −11.05149480962974125964177867677, −9.928918197062244174519750232199, −8.986276119359527707921251015141, −6.44304510033463415173484092572, −5.12446628284446948598835557297, −3.87408855994787623833612340101, −2.54866113577040171331492330518,
3.75731764435968774470892816384, 4.95204838226911371117602036292, 6.36847925704704372442647373315, 7.35232893365502883481237228043, 8.071843040739414971117056122167, 9.868619263792397491667947412371, 12.08286885366379150780480349447, 12.38603921139558699823896182455, 13.71309281537880333588503357547, 14.27616951443288338087874889876