L(s) = 1 | + (0.866 − 1.5i)3-s − 3.46i·5-s + (2.59 − 1.5i)7-s + (−4.33 − 2.5i)11-s + (−1 + 3.46i)13-s + (−5.19 − 2.99i)15-s + (3.5 + 6.06i)17-s + (−4.33 + 2.5i)19-s − 5.19i·21-s + (2.59 − 4.5i)23-s − 6.99·25-s + 5.19·27-s + (2.5 − 4.33i)29-s − 2i·31-s + (−7.5 + 4.33i)33-s + ⋯ |
L(s) = 1 | + (0.499 − 0.866i)3-s − 1.54i·5-s + (0.981 − 0.566i)7-s + (−1.30 − 0.753i)11-s + (−0.277 + 0.960i)13-s + (−1.34 − 0.774i)15-s + (0.848 + 1.47i)17-s + (−0.993 + 0.573i)19-s − 1.13i·21-s + (0.541 − 0.938i)23-s − 1.39·25-s + 1.00·27-s + (0.464 − 0.804i)29-s − 0.359i·31-s + (−1.30 + 0.753i)33-s + ⋯ |
Λ(s)=(=(416s/2ΓC(s)L(s)(−0.265+0.964i)Λ(2−s)
Λ(s)=(=(416s/2ΓC(s+1/2)L(s)(−0.265+0.964i)Λ(1−s)
Degree: |
2 |
Conductor: |
416
= 25⋅13
|
Sign: |
−0.265+0.964i
|
Analytic conductor: |
3.32177 |
Root analytic conductor: |
1.82257 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ416(225,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 416, ( :1/2), −0.265+0.964i)
|
Particular Values
L(1) |
≈ |
1.00432−1.31759i |
L(21) |
≈ |
1.00432−1.31759i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 13 | 1+(1−3.46i)T |
good | 3 | 1+(−0.866+1.5i)T+(−1.5−2.59i)T2 |
| 5 | 1+3.46iT−5T2 |
| 7 | 1+(−2.59+1.5i)T+(3.5−6.06i)T2 |
| 11 | 1+(4.33+2.5i)T+(5.5+9.52i)T2 |
| 17 | 1+(−3.5−6.06i)T+(−8.5+14.7i)T2 |
| 19 | 1+(4.33−2.5i)T+(9.5−16.4i)T2 |
| 23 | 1+(−2.59+4.5i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−2.5+4.33i)T+(−14.5−25.1i)T2 |
| 31 | 1+2iT−31T2 |
| 37 | 1+(−4.5−2.59i)T+(18.5+32.0i)T2 |
| 41 | 1+(1.5+0.866i)T+(20.5+35.5i)T2 |
| 43 | 1+(−2.59−4.5i)T+(−21.5+37.2i)T2 |
| 47 | 1+4iT−47T2 |
| 53 | 1−4T+53T2 |
| 59 | 1+(−6.06+3.5i)T+(29.5−51.0i)T2 |
| 61 | 1+(1.5+2.59i)T+(−30.5+52.8i)T2 |
| 67 | 1+(2.59+1.5i)T+(33.5+58.0i)T2 |
| 71 | 1+(6.06−3.5i)T+(35.5−61.4i)T2 |
| 73 | 1+3.46iT−73T2 |
| 79 | 1+3.46T+79T2 |
| 83 | 1−14iT−83T2 |
| 89 | 1+(−1.5−0.866i)T+(44.5+77.0i)T2 |
| 97 | 1+(−7.5+4.33i)T+(48.5−84.0i)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
−10.94908538175983714923386239358, −10.08894683545246689525724414725, −8.592667418902307145411061986442, −8.252334559306368302886744114880, −7.65184186003859798424435421020, −6.18570872055557231223196832515, −4.99253488121029589828855222896, −4.19300724258586834100372526043, −2.20001653520271263739376382071, −1.11471873287324465725549016299,
2.54894164571844234500184399434, 3.13280708837345839580850573153, 4.69551890972061568647975851003, 5.51169308262128402487868568562, 7.09480649625657389865806807161, 7.67431865783985467018022019400, 8.830804606247835746766407911564, 9.922392955102943269953430468258, 10.45289529232732521623226162730, 11.18730186715297796826659150158