L(s) = 1 | + (0.618 − 1.90i)3-s + (−2.42 + 1.76i)5-s + (1.07 + 3.29i)7-s + (−0.809 − 0.587i)9-s + (4.20 + 3.05i)13-s + (1.85 + 5.70i)15-s + (4.20 − 3.05i)17-s + (−1.07 + 3.29i)19-s + 6.92·21-s − 6·23-s + (1.23 − 3.80i)25-s + (3.23 − 2.35i)27-s + (1.60 + 4.94i)29-s + (1.61 + 1.17i)31-s + (−8.40 − 6.10i)35-s + ⋯ |
L(s) = 1 | + (0.356 − 1.09i)3-s + (−1.08 + 0.788i)5-s + (0.404 + 1.24i)7-s + (−0.269 − 0.195i)9-s + (1.16 + 0.847i)13-s + (0.478 + 1.47i)15-s + (1.01 − 0.740i)17-s + (−0.245 + 0.755i)19-s + 1.51·21-s − 1.25·23-s + (0.247 − 0.760i)25-s + (0.622 − 0.452i)27-s + (0.298 + 0.917i)29-s + (0.290 + 0.211i)31-s + (−1.42 − 1.03i)35-s + ⋯ |
Λ(s)=(=(484s/2ΓC(s)L(s)(0.898−0.437i)Λ(2−s)
Λ(s)=(=(484s/2ΓC(s+1/2)L(s)(0.898−0.437i)Λ(1−s)
Degree: |
2 |
Conductor: |
484
= 22⋅112
|
Sign: |
0.898−0.437i
|
Analytic conductor: |
3.86475 |
Root analytic conductor: |
1.96589 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ484(81,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 484, ( :1/2), 0.898−0.437i)
|
Particular Values
L(1) |
≈ |
1.37035+0.316062i |
L(21) |
≈ |
1.37035+0.316062i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 11 | 1 |
good | 3 | 1+(−0.618+1.90i)T+(−2.42−1.76i)T2 |
| 5 | 1+(2.42−1.76i)T+(1.54−4.75i)T2 |
| 7 | 1+(−1.07−3.29i)T+(−5.66+4.11i)T2 |
| 13 | 1+(−4.20−3.05i)T+(4.01+12.3i)T2 |
| 17 | 1+(−4.20+3.05i)T+(5.25−16.1i)T2 |
| 19 | 1+(1.07−3.29i)T+(−15.3−11.1i)T2 |
| 23 | 1+6T+23T2 |
| 29 | 1+(−1.60−4.94i)T+(−23.4+17.0i)T2 |
| 31 | 1+(−1.61−1.17i)T+(9.57+29.4i)T2 |
| 37 | 1+(−0.309−0.951i)T+(−29.9+21.7i)T2 |
| 41 | 1+(−1.60+4.94i)T+(−33.1−24.0i)T2 |
| 43 | 1+43T2 |
| 47 | 1+(−1.85+5.70i)T+(−38.0−27.6i)T2 |
| 53 | 1+(−2.42−1.76i)T+(16.3+50.4i)T2 |
| 59 | 1+(−47.7+34.6i)T2 |
| 61 | 1+(11.2−8.14i)T+(18.8−58.0i)T2 |
| 67 | 1+2T+67T2 |
| 71 | 1+(21.9−67.5i)T2 |
| 73 | 1+(−2.14−6.58i)T+(−59.0+42.9i)T2 |
| 79 | 1+(−2.80−2.03i)T+(24.4+75.1i)T2 |
| 83 | 1+(8.40−6.10i)T+(25.6−78.9i)T2 |
| 89 | 1−15T+89T2 |
| 97 | 1+(4.04+2.93i)T+(29.9+92.2i)T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.34539378763698768062730011466, −10.28406838221100963188543133413, −8.881666859338304905959545279564, −8.205611213768174939401269582332, −7.49158919757557723340443892674, −6.61930686444053648316443566911, −5.63592215650930187885469294907, −4.04007893770970147807582721753, −2.88227273536465605248164758697, −1.65481890429106589581046978217,
0.932800103831873987413416072526, 3.45634746713046901247808587247, 4.08461867066035985153050145492, 4.74379905838905488574864231847, 6.16115110607912616539623375535, 7.79788212990948084812626670681, 8.064613599416460806924910895324, 9.110364935955963222628214807770, 10.20153258707320922605263001091, 10.69722164062410384369575536918