L(s) = 1 | + (0.866 + 0.5i)2-s + (2.12 − 1.22i)3-s + (0.499 + 0.866i)4-s + (1.81 + 1.30i)5-s + 2.44·6-s + 0.999i·8-s + (1.49 − 2.59i)9-s + (0.917 + 2.03i)10-s + (−2.44 − 4.24i)11-s + (2.12 + 1.22i)12-s − 0.449i·13-s + (5.44 + 0.550i)15-s + (−0.5 + 0.866i)16-s + (−1.73 + i)17-s + (2.59 − 1.49i)18-s + (−3.22 + 5.58i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (1.22 − 0.707i)3-s + (0.249 + 0.433i)4-s + (0.811 + 0.584i)5-s + 0.999·6-s + 0.353i·8-s + (0.499 − 0.866i)9-s + (0.290 + 0.644i)10-s + (−0.738 − 1.27i)11-s + (0.612 + 0.353i)12-s − 0.124i·13-s + (1.40 + 0.142i)15-s + (−0.125 + 0.216i)16-s + (−0.420 + 0.242i)17-s + (0.612 − 0.353i)18-s + (−0.739 + 1.28i)19-s + ⋯ |
Λ(s)=(=(490s/2ΓC(s)L(s)(0.988−0.152i)Λ(2−s)
Λ(s)=(=(490s/2ΓC(s+1/2)L(s)(0.988−0.152i)Λ(1−s)
Degree: |
2 |
Conductor: |
490
= 2⋅5⋅72
|
Sign: |
0.988−0.152i
|
Analytic conductor: |
3.91266 |
Root analytic conductor: |
1.97804 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ490(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 490, ( :1/2), 0.988−0.152i)
|
Particular Values
L(1) |
≈ |
3.03720+0.232528i |
L(21) |
≈ |
3.03720+0.232528i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.866−0.5i)T |
| 5 | 1+(−1.81−1.30i)T |
| 7 | 1 |
good | 3 | 1+(−2.12+1.22i)T+(1.5−2.59i)T2 |
| 11 | 1+(2.44+4.24i)T+(−5.5+9.52i)T2 |
| 13 | 1+0.449iT−13T2 |
| 17 | 1+(1.73−i)T+(8.5−14.7i)T2 |
| 19 | 1+(3.22−5.58i)T+(−9.5−16.4i)T2 |
| 23 | 1+(5.97+3.44i)T+(11.5+19.9i)T2 |
| 29 | 1−2.89T+29T2 |
| 31 | 1+(0.449+0.778i)T+(−15.5+26.8i)T2 |
| 37 | 1+(1.73+i)T+(18.5+32.0i)T2 |
| 41 | 1−10.8T+41T2 |
| 43 | 1+8.89iT−43T2 |
| 47 | 1+(0.778+0.449i)T+(23.5+40.7i)T2 |
| 53 | 1+(−0.953+0.550i)T+(26.5−45.8i)T2 |
| 59 | 1+(−3.22−5.58i)T+(−29.5+51.0i)T2 |
| 61 | 1+(−4.22+7.31i)T+(−30.5−52.8i)T2 |
| 67 | 1+(6.92−4i)T+(33.5−58.0i)T2 |
| 71 | 1+10.8T+71T2 |
| 73 | 1+(5.97−3.44i)T+(36.5−63.2i)T2 |
| 79 | 1+(1.44−2.51i)T+(−39.5−68.4i)T2 |
| 83 | 1−2.44iT−83T2 |
| 89 | 1+(−5+8.66i)T+(−44.5−77.0i)T2 |
| 97 | 1−3.79iT−97T2 |
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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
−10.87187111266764844253248957589, −10.17248898514516029166577760095, −8.801780826255665347991819872010, −8.234571254139816032386461872954, −7.38306539175079733475263343069, −6.27940874187768038126870490055, −5.66851242582104951387829776818, −3.92974511960388575541158503133, −2.86166375789601004525030271409, −2.04334240994385253305281612155,
2.03138564069713966821863161942, 2.78507681104172995137174624243, 4.30935194868626491306859508180, 4.78687288580673638851256134378, 6.08718308715875780005479392281, 7.38201506433324097357726109028, 8.506055264383167743124963974440, 9.383234403523106990313758510424, 9.882742541477205159014513266743, 10.70725252302961327699263635097