L(s) = 1 | + (−5.41 − 3.12i)5-s + (3.74 + 6.49i)7-s + (6.17 − 3.56i)11-s + (−0.888 + 1.53i)13-s − 14.7i·17-s − 19.9·19-s + (20.8 + 12.0i)23-s + (7.03 + 12.1i)25-s + (−40.1 + 23.1i)29-s + (14.1 − 24.5i)31-s − 46.8i·35-s + 63.0·37-s + (28.0 + 16.1i)41-s + (−38.8 − 67.2i)43-s + (38.4 − 22.2i)47-s + ⋯ |
L(s) = 1 | + (−1.08 − 0.625i)5-s + (0.535 + 0.927i)7-s + (0.561 − 0.324i)11-s + (−0.0683 + 0.118i)13-s − 0.869i·17-s − 1.04·19-s + (0.907 + 0.523i)23-s + (0.281 + 0.487i)25-s + (−1.38 + 0.798i)29-s + (0.456 − 0.790i)31-s − 1.33i·35-s + 1.70·37-s + (0.683 + 0.394i)41-s + (−0.902 − 1.56i)43-s + (0.818 − 0.472i)47-s + ⋯ |
Λ(s)=(=(864s/2ΓC(s)L(s)(0.00681+0.999i)Λ(3−s)
Λ(s)=(=(864s/2ΓC(s+1)L(s)(0.00681+0.999i)Λ(1−s)
Degree: |
2 |
Conductor: |
864
= 25⋅33
|
Sign: |
0.00681+0.999i
|
Analytic conductor: |
23.5422 |
Root analytic conductor: |
4.85204 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ864(449,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 864, ( :1), 0.00681+0.999i)
|
Particular Values
L(23) |
≈ |
1.153674500 |
L(21) |
≈ |
1.153674500 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
good | 5 | 1+(5.41+3.12i)T+(12.5+21.6i)T2 |
| 7 | 1+(−3.74−6.49i)T+(−24.5+42.4i)T2 |
| 11 | 1+(−6.17+3.56i)T+(60.5−104.i)T2 |
| 13 | 1+(0.888−1.53i)T+(−84.5−146.i)T2 |
| 17 | 1+14.7iT−289T2 |
| 19 | 1+19.9T+361T2 |
| 23 | 1+(−20.8−12.0i)T+(264.5+458.i)T2 |
| 29 | 1+(40.1−23.1i)T+(420.5−728.i)T2 |
| 31 | 1+(−14.1+24.5i)T+(−480.5−832.i)T2 |
| 37 | 1−63.0T+1.36e3T2 |
| 41 | 1+(−28.0−16.1i)T+(840.5+1.45e3i)T2 |
| 43 | 1+(38.8+67.2i)T+(−924.5+1.60e3i)T2 |
| 47 | 1+(−38.4+22.2i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1+42.2iT−2.80e3T2 |
| 59 | 1+(93.8+54.2i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(25.3+43.9i)T+(−1.86e3+3.22e3i)T2 |
| 67 | 1+(−56.9+98.6i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1+85.2iT−5.04e3T2 |
| 73 | 1+94.5T+5.32e3T2 |
| 79 | 1+(35.6+61.6i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(−94.9+54.7i)T+(3.44e3−5.96e3i)T2 |
| 89 | 1−29.6iT−7.92e3T2 |
| 97 | 1+(62.4+108.i)T+(−4.70e3+8.14e3i)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
−9.401189470109292628820687261839, −8.937554204409081609251800949589, −8.116772085409392942076304542715, −7.39036863734613415949918597083, −6.23696632093471127446093203369, −5.18518665973496425989621451283, −4.41825035423542562569148223015, −3.37082204931410259401212129891, −1.97880239174633552939571573617, −0.43115209277748592674697589857,
1.19557973065638955832278293003, 2.78995674984954172418860747262, 4.14898601425745734951311813554, 4.31425002238231150547035620048, 5.97787443792533052069671039034, 6.93329730773723431503799343600, 7.60966810194426634776850898290, 8.271943078676271022435783222724, 9.327159091442859742338551437471, 10.45069650039876994834128327120