L(s) = 1 | + 0.306i·3-s − 2.60i·7-s + 2.90·9-s + 2.71·11-s − 0.503i·13-s + 5.94i·17-s + 5.72·19-s + 0.799·21-s − 1.28i·23-s + 1.81i·27-s − 2.11·29-s + 3.95·31-s + 0.831i·33-s + 0.825i·37-s + 0.154·39-s + ⋯ |
L(s) = 1 | + 0.176i·3-s − 0.985i·7-s + 0.968·9-s + 0.818·11-s − 0.139i·13-s + 1.44i·17-s + 1.31·19-s + 0.174·21-s − 0.268i·23-s + 0.348i·27-s − 0.392·29-s + 0.710·31-s + 0.144i·33-s + 0.135i·37-s + 0.0246·39-s + ⋯ |
Λ(s)=(=(4000s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(4000s/2ΓC(s+1/2)L(s)Λ(1−s)
Degree: |
2 |
Conductor: |
4000
= 25⋅53
|
Sign: |
1
|
Analytic conductor: |
31.9401 |
Root analytic conductor: |
5.65156 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ4000(1249,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 4000, ( :1/2), 1)
|
Particular Values
L(1) |
≈ |
2.341334368 |
L(21) |
≈ |
2.341334368 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
good | 3 | 1−0.306iT−3T2 |
| 7 | 1+2.60iT−7T2 |
| 11 | 1−2.71T+11T2 |
| 13 | 1+0.503iT−13T2 |
| 17 | 1−5.94iT−17T2 |
| 19 | 1−5.72T+19T2 |
| 23 | 1+1.28iT−23T2 |
| 29 | 1+2.11T+29T2 |
| 31 | 1−3.95T+31T2 |
| 37 | 1−0.825iT−37T2 |
| 41 | 1+4.53T+41T2 |
| 43 | 1−5.38iT−43T2 |
| 47 | 1−5.62iT−47T2 |
| 53 | 1+10.9iT−53T2 |
| 59 | 1−13.7T+59T2 |
| 61 | 1+7.00T+61T2 |
| 67 | 1−2.85iT−67T2 |
| 71 | 1+11.3T+71T2 |
| 73 | 1−10.1iT−73T2 |
| 79 | 1−11.1T+79T2 |
| 83 | 1+5.83iT−83T2 |
| 89 | 1+13.7T+89T2 |
| 97 | 1+7.02iT−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
−8.372969162505242410560318921969, −7.67164165729430461276154082386, −6.97639971187206771505235934338, −6.40396303678978662247444718085, −5.44473799146764295973591890567, −4.44928356940705091490909803987, −3.95298573379079286218799589903, −3.19689926592584631656563085779, −1.71107400391760168829292155130, −0.974295861166339251123190171086,
0.915256463997868912796970581395, 1.94516425331264553051481797436, 2.91203111632549802402398799734, 3.81524129493345317976325938946, 4.78937250188975092036891343517, 5.42842211325491932501383546643, 6.27867802716520724464260757809, 7.12983332746628057564067134505, 7.49405381665830138392612920156, 8.566918253227958342263826728344