L(s) = 1 | + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (−0.0475 − 0.998i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (1.16 − 0.600i)13-s + (−0.959 − 0.281i)16-s + (0.143 + 0.124i)19-s + (0.786 − 0.618i)21-s + (−0.888 + 0.458i)25-s + (−0.959 + 0.281i)27-s + (−0.995 − 0.0950i)28-s + (1.61 − 1.03i)31-s + (0.888 + 0.458i)36-s + 0.0951·37-s + ⋯ |
L(s) = 1 | + (0.580 + 0.814i)3-s + (0.142 − 0.989i)4-s + (−0.0475 − 0.998i)7-s + (−0.327 + 0.945i)9-s + (0.888 − 0.458i)12-s + (1.16 − 0.600i)13-s + (−0.959 − 0.281i)16-s + (0.143 + 0.124i)19-s + (0.786 − 0.618i)21-s + (−0.888 + 0.458i)25-s + (−0.959 + 0.281i)27-s + (−0.995 − 0.0950i)28-s + (1.61 − 1.03i)31-s + (0.888 + 0.458i)36-s + 0.0951·37-s + ⋯ |
Λ(s)=(=(1407s/2ΓC(s)L(s)(0.925+0.378i)Λ(1−s)
Λ(s)=(=(1407s/2ΓC(s)L(s)(0.925+0.378i)Λ(1−s)
Degree: |
2 |
Conductor: |
1407
= 3⋅7⋅67
|
Sign: |
0.925+0.378i
|
Analytic conductor: |
0.702184 |
Root analytic conductor: |
0.837964 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1407(1319,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1407, ( :0), 0.925+0.378i)
|
Particular Values
L(21) |
≈ |
1.365655437 |
L(21) |
≈ |
1.365655437 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(−0.580−0.814i)T |
| 7 | 1+(0.0475+0.998i)T |
| 67 | 1+(0.0475−0.998i)T |
good | 2 | 1+(−0.142+0.989i)T2 |
| 5 | 1+(0.888−0.458i)T2 |
| 11 | 1+(0.841+0.540i)T2 |
| 13 | 1+(−1.16+0.600i)T+(0.580−0.814i)T2 |
| 17 | 1+(0.235−0.971i)T2 |
| 19 | 1+(−0.143−0.124i)T+(0.142+0.989i)T2 |
| 23 | 1+(0.654+0.755i)T2 |
| 29 | 1+(0.5−0.866i)T2 |
| 31 | 1+(−1.61+1.03i)T+(0.415−0.909i)T2 |
| 37 | 1−0.0951T+T2 |
| 41 | 1+(−0.235+0.971i)T2 |
| 43 | 1+(−1.07+0.153i)T+(0.959−0.281i)T2 |
| 47 | 1+(−0.327+0.945i)T2 |
| 53 | 1+(0.235+0.971i)T2 |
| 59 | 1+(−0.995+0.0950i)T2 |
| 61 | 1+(1.91−0.560i)T+(0.841−0.540i)T2 |
| 71 | 1+(−0.723+0.690i)T2 |
| 73 | 1+(1.34−1.40i)T+(−0.0475−0.998i)T2 |
| 79 | 1+(−0.915−1.77i)T+(−0.580+0.814i)T2 |
| 83 | 1+(0.0475−0.998i)T2 |
| 89 | 1+(−0.327−0.945i)T2 |
| 97 | 1+(−0.580+1.00i)T+(−0.5−0.866i)T2 |
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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
−9.881015897810035803863165780698, −9.069101747719300265267699396625, −8.154262435546445845339174261133, −7.40663482207328874149709720429, −6.24031460469296276692130018724, −5.57078669819718937243750182116, −4.48278792555911361742751908304, −3.81845553794549652178629245485, −2.65940897839852724161540275496, −1.22831713221134968030026347555,
1.72166448917007778309369517735, 2.73564177481551775723597103750, 3.48452945053310921058703218116, 4.57877484337510394353819784972, 6.10405023411206937068844189145, 6.47394956410698699440949465613, 7.58762625485116832485378334498, 8.179104344986139503165865029420, 8.879569972089805198703996439288, 9.340092826008785885035860783214