L(s) = 1 | + 9i·3-s − 26i·7-s − 54·9-s + 59·11-s − 28i·13-s + 5i·17-s + 109·19-s + 234·21-s − 194i·23-s − 243i·27-s + 32·29-s − 10·31-s + 531i·33-s − 198i·37-s + 252·39-s + ⋯ |
L(s) = 1 | + 1.73i·3-s − 1.40i·7-s − 2·9-s + 1.61·11-s − 0.597i·13-s + 0.0713i·17-s + 1.31·19-s + 2.43·21-s − 1.75i·23-s − 1.73i·27-s + 0.204·29-s − 0.0579·31-s + 2.80i·33-s − 0.879i·37-s + 1.03·39-s + ⋯ |
Λ(s)=(=(400s/2ΓC(s)L(s)(0.894−0.447i)Λ(4−s)
Λ(s)=(=(400s/2ΓC(s+3/2)L(s)(0.894−0.447i)Λ(1−s)
Degree: |
2 |
Conductor: |
400
= 24⋅52
|
Sign: |
0.894−0.447i
|
Analytic conductor: |
23.6007 |
Root analytic conductor: |
4.85806 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ400(49,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 400, ( :3/2), 0.894−0.447i)
|
Particular Values
L(2) |
≈ |
1.983063971 |
L(21) |
≈ |
1.983063971 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
good | 3 | 1−9iT−27T2 |
| 7 | 1+26iT−343T2 |
| 11 | 1−59T+1.33e3T2 |
| 13 | 1+28iT−2.19e3T2 |
| 17 | 1−5iT−4.91e3T2 |
| 19 | 1−109T+6.85e3T2 |
| 23 | 1+194iT−1.21e4T2 |
| 29 | 1−32T+2.43e4T2 |
| 31 | 1+10T+2.97e4T2 |
| 37 | 1+198iT−5.06e4T2 |
| 41 | 1−117T+6.89e4T2 |
| 43 | 1−388iT−7.95e4T2 |
| 47 | 1−68iT−1.03e5T2 |
| 53 | 1−18iT−1.48e5T2 |
| 59 | 1−392T+2.05e5T2 |
| 61 | 1+710T+2.26e5T2 |
| 67 | 1−253iT−3.00e5T2 |
| 71 | 1−612T+3.57e5T2 |
| 73 | 1−549iT−3.89e5T2 |
| 79 | 1−414T+4.93e5T2 |
| 83 | 1+121iT−5.71e5T2 |
| 89 | 1−81T+7.04e5T2 |
| 97 | 1+1.50e3iT−9.12e5T2 |
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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
−10.79043075621965506230927611353, −9.987603211268030673018956314429, −9.399032736504924727903376186795, −8.407324895568196863055913111781, −7.15146438975704387116974152662, −5.99391113241072707724401966422, −4.67623808152103324851291646450, −4.05934519329582865181703773207, −3.18155000731107341229798617985, −0.812327148724330407414618151455,
1.18780125828249755626139227223, 2.07207612323681275808946182862, 3.40014802688379294242366373557, 5.34276772508495752974998814740, 6.19836486507960859299430145951, 6.96833061995850828091263918969, 7.87219773956820600970951985115, 8.951127617574279610919421900308, 9.434459075668451143500661635437, 11.38656219364851894048553306075