L(s) = 1 | − 2-s + (0.536 − 0.536i)3-s + 4-s + (−2.23 + 0.127i)5-s + (−0.536 + 0.536i)6-s + (0.767 − 0.767i)7-s − 8-s + 2.42i·9-s + (2.23 − 0.127i)10-s − 4.39i·11-s + (0.536 − 0.536i)12-s + 6.74·13-s + (−0.767 + 0.767i)14-s + (−1.12 + 1.26i)15-s + 16-s − 7.34i·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (0.309 − 0.309i)3-s + 0.5·4-s + (−0.998 + 0.0571i)5-s + (−0.219 + 0.219i)6-s + (0.290 − 0.290i)7-s − 0.353·8-s + 0.808i·9-s + (0.705 − 0.0404i)10-s − 1.32i·11-s + (0.154 − 0.154i)12-s + 1.87·13-s + (−0.205 + 0.205i)14-s + (−0.291 + 0.326i)15-s + 0.250·16-s − 1.78i·17-s + ⋯ |
Λ(s)=(=(370s/2ΓC(s)L(s)(0.458+0.888i)Λ(2−s)
Λ(s)=(=(370s/2ΓC(s+1/2)L(s)(0.458+0.888i)Λ(1−s)
Degree: |
2 |
Conductor: |
370
= 2⋅5⋅37
|
Sign: |
0.458+0.888i
|
Analytic conductor: |
2.95446 |
Root analytic conductor: |
1.71885 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ370(253,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 370, ( :1/2), 0.458+0.888i)
|
Particular Values
L(1) |
≈ |
0.802326−0.489132i |
L(21) |
≈ |
0.802326−0.489132i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+T |
| 5 | 1+(2.23−0.127i)T |
| 37 | 1+(−6.04+0.633i)T |
good | 3 | 1+(−0.536+0.536i)T−3iT2 |
| 7 | 1+(−0.767+0.767i)T−7iT2 |
| 11 | 1+4.39iT−11T2 |
| 13 | 1−6.74T+13T2 |
| 17 | 1+7.34iT−17T2 |
| 19 | 1+(2.59+2.59i)T+19iT2 |
| 23 | 1−1.20T+23T2 |
| 29 | 1+(1.25−1.25i)T−29iT2 |
| 31 | 1+(4.14+4.14i)T+31iT2 |
| 41 | 1+4.07iT−41T2 |
| 43 | 1+8.56T+43T2 |
| 47 | 1+(−7.68+7.68i)T−47iT2 |
| 53 | 1+(−2.31−2.31i)T+53iT2 |
| 59 | 1+(−7.61−7.61i)T+59iT2 |
| 61 | 1+(−1.14−1.14i)T+61iT2 |
| 67 | 1+(−6.25−6.25i)T+67iT2 |
| 71 | 1+5.46T+71T2 |
| 73 | 1+(1.88−1.88i)T−73iT2 |
| 79 | 1+(5.13+5.13i)T+79iT2 |
| 83 | 1+(−0.570−0.570i)T+83iT2 |
| 89 | 1+(7.54−7.54i)T−89iT2 |
| 97 | 1−5.39iT−97T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.15942386887266794319881225171, −10.63888344079404118892256147635, −8.976016077829635454927211228455, −8.497724961830795629062060478467, −7.65547178197472295655974375457, −6.84023544553937522234617642602, −5.48312056699927288382531036098, −3.96180408119657111913556146283, −2.76886333314809316531144963513, −0.865287218671018101826048689564,
1.57498701703147038738978322923, 3.50279409985366115401881304839, 4.25182232808081372223240095544, 6.01441385892366341713050377685, 6.94958425922972055595891682828, 8.307635307273956549335690547932, 8.486896700602345151557878900724, 9.651768416117621174174194272689, 10.63540929523805678505360724096, 11.36556820883056863407576363104