L(s) = 1 | + (1.41 − 0.0361i)2-s + (−2.77 + 1.60i)3-s + (1.99 − 0.102i)4-s + 5-s + (−3.87 + 2.36i)6-s + (1.83 + 1.06i)7-s + (2.82 − 0.216i)8-s + (3.64 − 6.31i)9-s + (1.41 − 0.0361i)10-s + (1.08 + 1.88i)11-s + (−5.38 + 3.48i)12-s + (−0.138 − 3.60i)13-s + (2.63 + 1.43i)14-s + (−2.77 + 1.60i)15-s + (3.97 − 0.407i)16-s + (−3.44 + 5.97i)17-s + ⋯ |
L(s) = 1 | + (0.999 − 0.0255i)2-s + (−1.60 + 0.926i)3-s + (0.998 − 0.0510i)4-s + 0.447·5-s + (−1.58 + 0.966i)6-s + (0.695 + 0.401i)7-s + (0.997 − 0.0765i)8-s + (1.21 − 2.10i)9-s + (0.447 − 0.0114i)10-s + (0.328 + 0.568i)11-s + (−1.55 + 1.00i)12-s + (−0.0385 − 0.999i)13-s + (0.705 + 0.383i)14-s + (−0.717 + 0.414i)15-s + (0.994 − 0.101i)16-s + (−0.836 + 1.44i)17-s + ⋯ |
Λ(s)=(=(520s/2ΓC(s)L(s)(0.408−0.912i)Λ(2−s)
Λ(s)=(=(520s/2ΓC(s+1/2)L(s)(0.408−0.912i)Λ(1−s)
Degree: |
2 |
Conductor: |
520
= 23⋅5⋅13
|
Sign: |
0.408−0.912i
|
Analytic conductor: |
4.15222 |
Root analytic conductor: |
2.03769 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ520(101,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 520, ( :1/2), 0.408−0.912i)
|
Particular Values
L(1) |
≈ |
1.60392+1.03887i |
L(21) |
≈ |
1.60392+1.03887i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−1.41+0.0361i)T |
| 5 | 1−T |
| 13 | 1+(0.138+3.60i)T |
good | 3 | 1+(2.77−1.60i)T+(1.5−2.59i)T2 |
| 7 | 1+(−1.83−1.06i)T+(3.5+6.06i)T2 |
| 11 | 1+(−1.08−1.88i)T+(−5.5+9.52i)T2 |
| 17 | 1+(3.44−5.97i)T+(−8.5−14.7i)T2 |
| 19 | 1+(1.91−3.30i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−3.82−6.63i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−1.73+1.00i)T+(14.5−25.1i)T2 |
| 31 | 1+7.18iT−31T2 |
| 37 | 1+(−2.98−5.16i)T+(−18.5+32.0i)T2 |
| 41 | 1+(1.20−0.694i)T+(20.5−35.5i)T2 |
| 43 | 1+(6.91+3.99i)T+(21.5+37.2i)T2 |
| 47 | 1+1.65iT−47T2 |
| 53 | 1+8.68iT−53T2 |
| 59 | 1+(−4.70+8.14i)T+(−29.5−51.0i)T2 |
| 61 | 1+(0.984+0.568i)T+(30.5+52.8i)T2 |
| 67 | 1+(−4.36−7.55i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−0.113−0.0655i)T+(35.5+61.4i)T2 |
| 73 | 1−3.75iT−73T2 |
| 79 | 1+11.3T+79T2 |
| 83 | 1+11.5T+83T2 |
| 89 | 1+(−11.9+6.89i)T+(44.5−77.0i)T2 |
| 97 | 1+(−0.694−0.400i)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
−11.33521552723940323353287311888, −10.31836500760293413959506495788, −9.866370606117120000299570847472, −8.287576067553610957270472771307, −6.86256690928652443756448637075, −5.99349772912387576106233753578, −5.41136870771025240743972525159, −4.60133243247408464192148651683, −3.67088096509626606233666429729, −1.70933462277398731718714744397,
1.13420540021847292407329481102, 2.43973080490837302899662171258, 4.63310247181587119922070155702, 4.90955825187276654290819855878, 6.18230956489290559517201579179, 6.76610689811139623638332370579, 7.34793509026718400454116334498, 8.832180528683623896362760018171, 10.47346351703949479980330432415, 11.14205118014245319640860163496