L(s) = 1 | + (−0.5 − 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (0.292 + 0.507i)5-s − 0.999·6-s + (1.62 − 2.09i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + (0.292 − 0.507i)10-s + (0.207 − 0.358i)11-s + (0.499 + 0.866i)12-s + 13-s + (−2.62 − 0.358i)14-s + 0.585·15-s + (−0.5 − 0.866i)16-s + (1.20 − 2.09i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.130 + 0.226i)5-s − 0.408·6-s + (0.612 − 0.790i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + (0.0926 − 0.160i)10-s + (0.0624 − 0.108i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (−0.700 − 0.0958i)14-s + 0.151·15-s + (−0.125 − 0.216i)16-s + (0.292 − 0.507i)17-s + ⋯ |
Λ(s)=(=(546s/2ΓC(s)L(s)(−0.198+0.980i)Λ(2−s)
Λ(s)=(=(546s/2ΓC(s+1/2)L(s)(−0.198+0.980i)Λ(1−s)
Degree: |
2 |
Conductor: |
546
= 2⋅3⋅7⋅13
|
Sign: |
−0.198+0.980i
|
Analytic conductor: |
4.35983 |
Root analytic conductor: |
2.08802 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ546(235,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 546, ( :1/2), −0.198+0.980i)
|
Particular Values
L(1) |
≈ |
0.879040−1.07442i |
L(21) |
≈ |
0.879040−1.07442i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.5+0.866i)T |
| 3 | 1+(−0.5+0.866i)T |
| 7 | 1+(−1.62+2.09i)T |
| 13 | 1−T |
good | 5 | 1+(−0.292−0.507i)T+(−2.5+4.33i)T2 |
| 11 | 1+(−0.207+0.358i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−1.20+2.09i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−1.08−1.88i)T+(−9.5+16.4i)T2 |
| 23 | 1+(0.707+1.22i)T+(−11.5+19.9i)T2 |
| 29 | 1+1.82T+29T2 |
| 31 | 1+(−4.24+7.34i)T+(−15.5−26.8i)T2 |
| 37 | 1+(0.707+1.22i)T+(−18.5+32.0i)T2 |
| 41 | 1+9.89T+41T2 |
| 43 | 1−6.48T+43T2 |
| 47 | 1+(0.5+0.866i)T+(−23.5+40.7i)T2 |
| 53 | 1+(4.74−8.21i)T+(−26.5−45.8i)T2 |
| 59 | 1+(1.03−1.79i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−2.20−3.82i)T+(−30.5+52.8i)T2 |
| 67 | 1+(0.914−1.58i)T+(−33.5−58.0i)T2 |
| 71 | 1+5T+71T2 |
| 73 | 1+(0.707−1.22i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−5.82−10.0i)T+(−39.5+68.4i)T2 |
| 83 | 1−7.65T+83T2 |
| 89 | 1+(1.29+2.23i)T+(−44.5+77.0i)T2 |
| 97 | 1+0.928T+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
−10.53385908097201773531985822929, −9.821660577831951120051322361002, −8.743223353260170701923370684849, −7.925004841112605835894633203342, −7.21376738959451813231329187334, −6.07930762479780528773528171552, −4.65417418864662759117382505018, −3.55997991111337207290095287005, −2.31390321768202636331679930620, −0.976092173353726931486961429346,
1.67762116461371694294480969108, 3.29016691513859500218327276100, 4.74854054699141955874356324643, 5.40646758245438806626184321245, 6.48963198252313643067441513115, 7.66713824565087876907251207664, 8.533040214341072666452457228795, 9.070583009165941039957521976852, 9.981024876351206593514317371916, 10.89060273090623006307978306006