L(s) = 1 | + (−0.811 − 0.318i)2-s + (−0.908 − 0.843i)4-s + (0.228 + 0.155i)5-s + (−2.58 + 0.584i)7-s + (1.22 + 2.54i)8-s + (−0.135 − 0.199i)10-s + (0.640 + 4.25i)11-s + (3.73 − 2.98i)13-s + (2.28 + 0.347i)14-s + (0.00131 + 0.0176i)16-s + (5.10 + 1.57i)17-s + (1.61 − 0.930i)19-s + (−0.0764 − 0.334i)20-s + (0.833 − 3.65i)22-s + (0.312 + 1.01i)23-s + ⋯ |
L(s) = 1 | + (−0.573 − 0.225i)2-s + (−0.454 − 0.421i)4-s + (0.102 + 0.0697i)5-s + (−0.975 + 0.220i)7-s + (0.433 + 0.899i)8-s + (−0.0429 − 0.0630i)10-s + (0.193 + 1.28i)11-s + (1.03 − 0.826i)13-s + (0.609 + 0.0928i)14-s + (0.000329 + 0.00440i)16-s + (1.23 + 0.381i)17-s + (0.369 − 0.213i)19-s + (−0.0170 − 0.0748i)20-s + (0.177 − 0.778i)22-s + (0.0652 + 0.211i)23-s + ⋯ |
Λ(s)=(=(441s/2ΓC(s)L(s)(0.994−0.107i)Λ(2−s)
Λ(s)=(=(441s/2ΓC(s+1/2)L(s)(0.994−0.107i)Λ(1−s)
Degree: |
2 |
Conductor: |
441
= 32⋅72
|
Sign: |
0.994−0.107i
|
Analytic conductor: |
3.52140 |
Root analytic conductor: |
1.87654 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ441(395,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 441, ( :1/2), 0.994−0.107i)
|
Particular Values
L(1) |
≈ |
0.868198+0.0468744i |
L(21) |
≈ |
0.868198+0.0468744i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 7 | 1+(2.58−0.584i)T |
good | 2 | 1+(0.811+0.318i)T+(1.46+1.36i)T2 |
| 5 | 1+(−0.228−0.155i)T+(1.82+4.65i)T2 |
| 11 | 1+(−0.640−4.25i)T+(−10.5+3.24i)T2 |
| 13 | 1+(−3.73+2.98i)T+(2.89−12.6i)T2 |
| 17 | 1+(−5.10−1.57i)T+(14.0+9.57i)T2 |
| 19 | 1+(−1.61+0.930i)T+(9.5−16.4i)T2 |
| 23 | 1+(−0.312−1.01i)T+(−19.0+12.9i)T2 |
| 29 | 1+(−3.23+0.737i)T+(26.1−12.5i)T2 |
| 31 | 1+(−4.59−2.65i)T+(15.5+26.8i)T2 |
| 37 | 1+(5.88−5.46i)T+(2.76−36.8i)T2 |
| 41 | 1+(−5.58+2.69i)T+(25.5−32.0i)T2 |
| 43 | 1+(−9.74−4.69i)T+(26.8+33.6i)T2 |
| 47 | 1+(3.78−9.64i)T+(−34.4−31.9i)T2 |
| 53 | 1+(−2.18+2.35i)T+(−3.96−52.8i)T2 |
| 59 | 1+(10.0−6.84i)T+(21.5−54.9i)T2 |
| 61 | 1+(−5.45−5.88i)T+(−4.55+60.8i)T2 |
| 67 | 1+(−3.64+6.32i)T+(−33.5−58.0i)T2 |
| 71 | 1+(−4.23−0.965i)T+(63.9+30.8i)T2 |
| 73 | 1+(4.31−1.69i)T+(53.5−49.6i)T2 |
| 79 | 1+(−1.86−3.22i)T+(−39.5+68.4i)T2 |
| 83 | 1+(0.641−0.804i)T+(−18.4−80.9i)T2 |
| 89 | 1+(15.0+2.26i)T+(85.0+26.2i)T2 |
| 97 | 1+10.6iT−97T2 |
show more | |
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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
−10.78008779532617352696913766378, −10.04422836175059224210413620469, −9.590670012914396068280698976795, −8.563479747477411679496765387291, −7.67219915712716445344812076840, −6.36550230805970515800851718314, −5.53180375957216189832039643559, −4.27175242162663013624359366580, −2.86890127153711277467523019838, −1.18467220126356078468143564374,
0.868979418544816973011671512625, 3.26479514125089488407130257666, 3.94095866103809508792347013594, 5.58973264812282173132921054867, 6.54666230713029842391459552170, 7.53799795256729566962859136435, 8.529753418129927530529148365241, 9.217753878757298983625578971447, 9.943626441680640906544462483993, 10.98219075929003445428115648680