L(s) = 1 | − 1.51·2-s + (−1.66 − 0.476i)3-s + 0.294·4-s + (0.775 − 2.09i)5-s + (2.52 + 0.722i)6-s + 2.58·8-s + (2.54 + 1.58i)9-s + (−1.17 + 3.17i)10-s + 2.14i·11-s + (−0.490 − 0.140i)12-s − 3.48·13-s + (−2.29 + 3.12i)15-s − 4.50·16-s + 3.57i·17-s + (−3.85 − 2.40i)18-s − 1.22i·19-s + ⋯ |
L(s) = 1 | − 1.07·2-s + (−0.961 − 0.275i)3-s + 0.147·4-s + (0.346 − 0.937i)5-s + (1.02 + 0.294i)6-s + 0.913·8-s + (0.848 + 0.529i)9-s + (−0.371 + 1.00i)10-s + 0.647i·11-s + (−0.141 − 0.0405i)12-s − 0.965·13-s + (−0.591 + 0.806i)15-s − 1.12·16-s + 0.867i·17-s + (−0.908 − 0.566i)18-s − 0.280i·19-s + ⋯ |
Λ(s)=(=(735s/2ΓC(s)L(s)(0.975−0.221i)Λ(2−s)
Λ(s)=(=(735s/2ΓC(s+1/2)L(s)(0.975−0.221i)Λ(1−s)
Degree: |
2 |
Conductor: |
735
= 3⋅5⋅72
|
Sign: |
0.975−0.221i
|
Analytic conductor: |
5.86900 |
Root analytic conductor: |
2.42260 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ735(734,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 735, ( :1/2), 0.975−0.221i)
|
Particular Values
L(1) |
≈ |
0.500436+0.0562466i |
L(21) |
≈ |
0.500436+0.0562466i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.66+0.476i)T |
| 5 | 1+(−0.775+2.09i)T |
| 7 | 1 |
good | 2 | 1+1.51T+2T2 |
| 11 | 1−2.14iT−11T2 |
| 13 | 1+3.48T+13T2 |
| 17 | 1−3.57iT−17T2 |
| 19 | 1+1.22iT−19T2 |
| 23 | 1−1.51T+23T2 |
| 29 | 1−5.95iT−29T2 |
| 31 | 1+3.17iT−31T2 |
| 37 | 1−7.80iT−37T2 |
| 41 | 1−11.8T+41T2 |
| 43 | 1−2.99iT−43T2 |
| 47 | 1−6.10iT−47T2 |
| 53 | 1−11.2T+53T2 |
| 59 | 1+2.16T+59T2 |
| 61 | 1+3.39iT−61T2 |
| 67 | 1−10.3iT−67T2 |
| 71 | 1+10.3iT−71T2 |
| 73 | 1−6.85T+73T2 |
| 79 | 1+1.88T+79T2 |
| 83 | 1+9.10iT−83T2 |
| 89 | 1−1.77T+89T2 |
| 97 | 1−1.32T+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
−10.20195822778352397853823661247, −9.630888394188933091757836218470, −8.826538605411777022506774960746, −7.83986423520135101495408966361, −7.16334546629806346446861757023, −6.03218304013966704791314865584, −4.94303605644368709404316992082, −4.40253254363522616379739570003, −2.02674578096365277978578099842, −0.934157258717622031828074566707,
0.58413804265747357271270940679, 2.33943488734371595700592440606, 3.90642701935412226777154981957, 5.09101692645056914478478046596, 6.00054361514303294328229248228, 7.09619458023411752185073348117, 7.57143344896631631514888219502, 8.918102659414848953927303949689, 9.686775526056532084350634735722, 10.23542909689979247025013858767