L(s) = 1 | + (−1.69 − 0.353i)3-s + (0.469 + 0.812i)5-s + (2.75 + 1.19i)9-s + (−1.31 + 2.28i)11-s + (−2.71 − 4.69i)13-s + (−0.508 − 1.54i)15-s + 3.85·17-s − 1.09·19-s + (3.16 + 5.48i)23-s + (2.05 − 3.56i)25-s + (−4.24 − 3.00i)27-s + (−1.94 + 3.36i)29-s + (2.33 + 4.04i)31-s + (3.04 − 3.40i)33-s + 2.30·37-s + ⋯ |
L(s) = 1 | + (−0.979 − 0.203i)3-s + (0.209 + 0.363i)5-s + (0.916 + 0.399i)9-s + (−0.397 + 0.688i)11-s + (−0.751 − 1.30i)13-s + (−0.131 − 0.398i)15-s + 0.934·17-s − 0.251·19-s + (0.660 + 1.14i)23-s + (0.411 − 0.713i)25-s + (−0.816 − 0.577i)27-s + (−0.360 + 0.624i)29-s + (0.419 + 0.725i)31-s + (0.529 − 0.592i)33-s + 0.379·37-s + ⋯ |
Λ(s)=(=(1764s/2ΓC(s)L(s)(0.552−0.833i)Λ(2−s)
Λ(s)=(=(1764s/2ΓC(s+1/2)L(s)(0.552−0.833i)Λ(1−s)
Degree: |
2 |
Conductor: |
1764
= 22⋅32⋅72
|
Sign: |
0.552−0.833i
|
Analytic conductor: |
14.0856 |
Root analytic conductor: |
3.75308 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ1764(1177,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 1764, ( :1/2), 0.552−0.833i)
|
Particular Values
L(1) |
≈ |
1.050790975 |
L(21) |
≈ |
1.050790975 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+(1.69+0.353i)T |
| 7 | 1 |
good | 5 | 1+(−0.469−0.812i)T+(−2.5+4.33i)T2 |
| 11 | 1+(1.31−2.28i)T+(−5.5−9.52i)T2 |
| 13 | 1+(2.71+4.69i)T+(−6.5+11.2i)T2 |
| 17 | 1−3.85T+17T2 |
| 19 | 1+1.09T+19T2 |
| 23 | 1+(−3.16−5.48i)T+(−11.5+19.9i)T2 |
| 29 | 1+(1.94−3.36i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−2.33−4.04i)T+(−15.5+26.8i)T2 |
| 37 | 1−2.30T+37T2 |
| 41 | 1+(4.12+7.13i)T+(−20.5+35.5i)T2 |
| 43 | 1+(2.14−3.71i)T+(−21.5−37.2i)T2 |
| 47 | 1+(−1.32+2.28i)T+(−23.5−40.7i)T2 |
| 53 | 1+1.27T+53T2 |
| 59 | 1+(−3.02−5.23i)T+(−29.5+51.0i)T2 |
| 61 | 1+(6.71−11.6i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−3.64−6.31i)T+(−33.5+58.0i)T2 |
| 71 | 1−14.7T+71T2 |
| 73 | 1+5.74T+73T2 |
| 79 | 1+(5.51−9.54i)T+(−39.5−68.4i)T2 |
| 83 | 1+(1.24−2.15i)T+(−41.5−71.8i)T2 |
| 89 | 1−13.5T+89T2 |
| 97 | 1+(−1.75+3.03i)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
−9.811519132497296051682152415322, −8.549960275701370528970562850931, −7.46267648757675554983723052898, −7.22012646043601962713198896625, −6.12155861171008315406860100431, −5.33000211200466058999190658494, −4.83492789244022349656478981932, −3.47090339652815279905726140302, −2.38997450012751108376288751538, −1.03223110847777616672638492261,
0.54825955076921349334448080206, 1.89830263167812687633399299272, 3.29662638909496176090765054257, 4.50540588808464240856194354470, 5.00526120864087576722875937199, 5.96017072413000915988877392987, 6.61398850633789489292771503573, 7.48695886743064120524452440667, 8.427066630213209039497481256198, 9.413597974143889936702050471757