L(s) = 1 | − i·2-s + 0.302i·3-s − 4-s + 0.302·6-s + 4.60i·7-s + i·8-s + 2.90·9-s + 1.30·11-s − 0.302i·12-s + 2.30i·13-s + 4.60·14-s + 16-s − 6i·17-s − 2.90i·18-s − 2·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + 0.174i·3-s − 0.5·4-s + 0.123·6-s + 1.74i·7-s + 0.353i·8-s + 0.969·9-s + 0.392·11-s − 0.0874i·12-s + 0.638i·13-s + 1.23·14-s + 0.250·16-s − 1.45i·17-s − 0.685i·18-s − 0.458·19-s + ⋯ |
Λ(s)=(=(1850s/2ΓC(s)L(s)(0.447−0.894i)Λ(2−s)
Λ(s)=(=(1850s/2ΓC(s+1/2)L(s)(0.447−0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
1850
= 2⋅52⋅37
|
Sign: |
0.447−0.894i
|
Analytic conductor: |
14.7723 |
Root analytic conductor: |
3.84347 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1850(149,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1850, ( :1/2), 0.447−0.894i)
|
Particular Values
L(1) |
≈ |
1.471759425 |
L(21) |
≈ |
1.471759425 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+iT |
| 5 | 1 |
| 37 | 1−iT |
good | 3 | 1−0.302iT−3T2 |
| 7 | 1−4.60iT−7T2 |
| 11 | 1−1.30T+11T2 |
| 13 | 1−2.30iT−13T2 |
| 17 | 1+6iT−17T2 |
| 19 | 1+2T+19T2 |
| 23 | 1−6.90iT−23T2 |
| 29 | 1+6.90T+29T2 |
| 31 | 1−3.30T+31T2 |
| 41 | 1+0.908T+41T2 |
| 43 | 1−6.60iT−43T2 |
| 47 | 1+2.60iT−47T2 |
| 53 | 1−6iT−53T2 |
| 59 | 1+3.39T+59T2 |
| 61 | 1+10.5T+61T2 |
| 67 | 1−14.5iT−67T2 |
| 71 | 1−6T+71T2 |
| 73 | 1−8.69iT−73T2 |
| 79 | 1−16.1T+79T2 |
| 83 | 1+17.2iT−83T2 |
| 89 | 1+5.21T+89T2 |
| 97 | 1−12.4iT−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
−9.261035256779632435098519895436, −9.107242808474755464919339876143, −7.939258702848124228975144676821, −7.04342717995339147413024972463, −6.02735618369702835618610925151, −5.19516768060739279086933940796, −4.43809123102382197185587370236, −3.37000955572305406815820992691, −2.39197781983994159688185366103, −1.49857958825234531364959375963,
0.56774665187326436386963814195, 1.74670200477888196012967904422, 3.62079778364276613116429032460, 4.12974595855885795438674133190, 4.93171158631216274998438129249, 6.30087272990105668433948560233, 6.64808339352429282709096033937, 7.61022286175580840424156729173, 7.975997597319450431210831673884, 8.986845319187496416613010018571