L(s) = 1 | + (−0.229 + 4.99i)5-s − 8.73i·7-s + 10.4i·11-s + 5.38i·13-s − 26.2·17-s + 2.70·19-s + 33.2·23-s + (−24.8 − 2.28i)25-s + 17.4i·29-s − 48.3·31-s + (43.6 + 2.00i)35-s + 66.2i·37-s + 14.7i·41-s − 28.4i·43-s − 35.9·47-s + ⋯ |
L(s) = 1 | + (−0.0458 + 0.998i)5-s − 1.24i·7-s + 0.953i·11-s + 0.414i·13-s − 1.54·17-s + 0.142·19-s + 1.44·23-s + (−0.995 − 0.0915i)25-s + 0.602i·29-s − 1.56·31-s + (1.24 + 0.0572i)35-s + 1.79i·37-s + 0.359i·41-s − 0.661i·43-s − 0.764·47-s + ⋯ |
Λ(s)=(=(720s/2ΓC(s)L(s)(−0.789−0.614i)Λ(3−s)
Λ(s)=(=(720s/2ΓC(s+1)L(s)(−0.789−0.614i)Λ(1−s)
Degree: |
2 |
Conductor: |
720
= 24⋅32⋅5
|
Sign: |
−0.789−0.614i
|
Analytic conductor: |
19.6185 |
Root analytic conductor: |
4.42928 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ720(449,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 720, ( :1), −0.789−0.614i)
|
Particular Values
L(23) |
≈ |
0.8099794302 |
L(21) |
≈ |
0.8099794302 |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+(0.229−4.99i)T |
good | 7 | 1+8.73iT−49T2 |
| 11 | 1−10.4iT−121T2 |
| 13 | 1−5.38iT−169T2 |
| 17 | 1+26.2T+289T2 |
| 19 | 1−2.70T+361T2 |
| 23 | 1−33.2T+529T2 |
| 29 | 1−17.4iT−841T2 |
| 31 | 1+48.3T+961T2 |
| 37 | 1−66.2iT−1.36e3T2 |
| 41 | 1−14.7iT−1.68e3T2 |
| 43 | 1+28.4iT−1.84e3T2 |
| 47 | 1+35.9T+2.20e3T2 |
| 53 | 1+42.2T+2.80e3T2 |
| 59 | 1−55.9iT−3.48e3T2 |
| 61 | 1+96.1T+3.72e3T2 |
| 67 | 1+15.4iT−4.48e3T2 |
| 71 | 1−13.5iT−5.04e3T2 |
| 73 | 1+63.7iT−5.32e3T2 |
| 79 | 1+94.5T+6.24e3T2 |
| 83 | 1+19.4T+6.88e3T2 |
| 89 | 1−118.iT−7.92e3T2 |
| 97 | 1−100.iT−9.40e3T2 |
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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
−10.76316992931983002128596702515, −9.788507030390641063362000267875, −8.965003834294115127766050304295, −7.65409311692330677360564191198, −6.97792499476209942165435871772, −6.54593790732215082874910037739, −4.93093544432469243043433512689, −4.09255683325102169790921487172, −3.00299431767854170063669968262, −1.65235532241044813063405729625,
0.27247822111649893783271917147, 1.89967053312408122838757118346, 3.13063080170806990179482072234, 4.47164843934817372484679825373, 5.43809620730264322216964866546, 6.03381436316480370647885244354, 7.32998054463873230889570172114, 8.499594704393535833607690882749, 8.888990430948075740132134356781, 9.540109463482324561324726485358