L(s) = 1 | + (1.09 − 0.497i)3-s + (−0.755 + 0.654i)5-s + (1.64 + 1.05i)7-s + (0.285 − 0.329i)9-s + (−0.497 + 1.09i)15-s + (2.31 + 0.333i)21-s + (0.599 + 0.800i)23-s + (0.142 − 0.989i)25-s + (−0.190 + 0.647i)27-s + (−1.25 + 0.368i)29-s + (−1.93 + 0.278i)35-s + (−0.708 − 0.817i)41-s + (−0.729 − 1.59i)43-s + 0.436i·45-s + 0.142i·47-s + ⋯ |
L(s) = 1 | + (1.09 − 0.497i)3-s + (−0.755 + 0.654i)5-s + (1.64 + 1.05i)7-s + (0.285 − 0.329i)9-s + (−0.497 + 1.09i)15-s + (2.31 + 0.333i)21-s + (0.599 + 0.800i)23-s + (0.142 − 0.989i)25-s + (−0.190 + 0.647i)27-s + (−1.25 + 0.368i)29-s + (−1.93 + 0.278i)35-s + (−0.708 − 0.817i)41-s + (−0.729 − 1.59i)43-s + 0.436i·45-s + 0.142i·47-s + ⋯ |
Λ(s)=(=(3680s/2ΓC(s)L(s)(0.808−0.588i)Λ(1−s)
Λ(s)=(=(3680s/2ΓC(s)L(s)(0.808−0.588i)Λ(1−s)
Degree: |
2 |
Conductor: |
3680
= 25⋅5⋅23
|
Sign: |
0.808−0.588i
|
Analytic conductor: |
1.83655 |
Root analytic conductor: |
1.35519 |
Motivic weight: |
0 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3680(3649,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 3680, ( :0), 0.808−0.588i)
|
Particular Values
L(21) |
≈ |
1.906115632 |
L(21) |
≈ |
1.906115632 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1+(0.755−0.654i)T |
| 23 | 1+(−0.599−0.800i)T |
good | 3 | 1+(−1.09+0.497i)T+(0.654−0.755i)T2 |
| 7 | 1+(−1.64−1.05i)T+(0.415+0.909i)T2 |
| 11 | 1+(0.959−0.281i)T2 |
| 13 | 1+(−0.415+0.909i)T2 |
| 17 | 1+(0.841−0.540i)T2 |
| 19 | 1+(−0.841−0.540i)T2 |
| 29 | 1+(1.25−0.368i)T+(0.841−0.540i)T2 |
| 31 | 1+(−0.654−0.755i)T2 |
| 37 | 1+(−0.142−0.989i)T2 |
| 41 | 1+(0.708+0.817i)T+(−0.142+0.989i)T2 |
| 43 | 1+(0.729+1.59i)T+(−0.654+0.755i)T2 |
| 47 | 1−0.142iT−T2 |
| 53 | 1+(0.415+0.909i)T2 |
| 59 | 1+(0.415−0.909i)T2 |
| 61 | 1+(−1.74−0.797i)T+(0.654+0.755i)T2 |
| 67 | 1+(−0.0994+0.691i)T+(−0.959−0.281i)T2 |
| 71 | 1+(−0.959−0.281i)T2 |
| 73 | 1+(−0.841−0.540i)T2 |
| 79 | 1+(−0.415+0.909i)T2 |
| 83 | 1+(1.22−1.41i)T+(−0.142−0.989i)T2 |
| 89 | 1+(−1.80+0.822i)T+(0.654−0.755i)T2 |
| 97 | 1+(−0.142+0.989i)T2 |
show more | |
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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
−8.650954471839809099903837315370, −8.086852005555389594001419695601, −7.44592784427851121662415834535, −6.98048495709471603173783506390, −5.63567856610402714854916817719, −5.09143248982754611236212811115, −3.97195211751783610767450591377, −3.17398191815913466890304235639, −2.27681551071833359156465535114, −1.66236203291104878264864555739,
1.05224602767372644078808664114, 2.12630305605077542984929108322, 3.37463734937184264094208059405, 4.02743116300192030575928637650, 4.65590458789864105735356932556, 5.21878626311066504081244890958, 6.62661397917501052559172023170, 7.62194398943483377381729460108, 7.947537552807281787971267507959, 8.529505867260315527596468832973