L(s) = 1 | + (0.707 + 0.707i)2-s + (0.292 + 0.292i)3-s + 1.00i·4-s + (0.707 + 2.12i)5-s + 0.414i·6-s + (−2.12 − 2.12i)7-s + (−0.707 + 0.707i)8-s − 2.82i·9-s + (−0.999 + 2i)10-s + (1.41 + 3i)11-s + (−0.292 + 0.292i)12-s + (3 − 3i)13-s − 3i·14-s + (−0.414 + 0.828i)15-s − 1.00·16-s + (−0.878 − 0.878i)17-s + ⋯ |
L(s) = 1 | + (0.499 + 0.499i)2-s + (0.169 + 0.169i)3-s + 0.500i·4-s + (0.316 + 0.948i)5-s + 0.169i·6-s + (−0.801 − 0.801i)7-s + (−0.250 + 0.250i)8-s − 0.942i·9-s + (−0.316 + 0.632i)10-s + (0.426 + 0.904i)11-s + (−0.0845 + 0.0845i)12-s + (0.832 − 0.832i)13-s − 0.801i·14-s + (−0.106 + 0.213i)15-s − 0.250·16-s + (−0.213 − 0.213i)17-s + ⋯ |
Λ(s)=(=(110s/2ΓC(s)L(s)(0.545−0.838i)Λ(2−s)
Λ(s)=(=(110s/2ΓC(s+1/2)L(s)(0.545−0.838i)Λ(1−s)
Degree: |
2 |
Conductor: |
110
= 2⋅5⋅11
|
Sign: |
0.545−0.838i
|
Analytic conductor: |
0.878354 |
Root analytic conductor: |
0.937205 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ110(43,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 110, ( :1/2), 0.545−0.838i)
|
Particular Values
L(1) |
≈ |
1.17909+0.639621i |
L(21) |
≈ |
1.17909+0.639621i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(−0.707−0.707i)T |
| 5 | 1+(−0.707−2.12i)T |
| 11 | 1+(−1.41−3i)T |
good | 3 | 1+(−0.292−0.292i)T+3iT2 |
| 7 | 1+(2.12+2.12i)T+7iT2 |
| 13 | 1+(−3+3i)T−13iT2 |
| 17 | 1+(0.878+0.878i)T+17iT2 |
| 19 | 1+3T+19T2 |
| 23 | 1+(5.82+5.82i)T+23iT2 |
| 29 | 1−7.24T+29T2 |
| 31 | 1−1.24T+31T2 |
| 37 | 1+(4.12−4.12i)T−37iT2 |
| 41 | 1−10.2iT−41T2 |
| 43 | 1+(7.24−7.24i)T−43iT2 |
| 47 | 1+(−1.58+1.58i)T−47iT2 |
| 53 | 1+(−2.46−2.46i)T+53iT2 |
| 59 | 1−1.41iT−59T2 |
| 61 | 1+1.24iT−61T2 |
| 67 | 1+(4−4i)T−67iT2 |
| 71 | 1−7.24T+71T2 |
| 73 | 1+(−6+6i)T−73iT2 |
| 79 | 1−1.75T+79T2 |
| 83 | 1+(−1.24+1.24i)T−83iT2 |
| 89 | 1−11.4iT−89T2 |
| 97 | 1+(−6.24+6.24i)T−97iT2 |
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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
−13.94899837191289466138790556587, −13.01695573167967684617661521812, −11.97009170806135149141438759919, −10.47587401765433130363582167783, −9.765061844068366462901407380717, −8.234419204157333859961660367458, −6.66982817656714865729393670030, −6.38135762585198493384827002430, −4.24691816132634715133386340351, −3.09814769631830291355157683830,
2.00281678291443671113746900445, 3.85074730143178893186960108990, 5.41647617266707158627460822550, 6.37611975826394405004313110237, 8.422900226516719833074242586972, 9.128431095279566032597316304171, 10.42010946134101646885186404052, 11.69325087857158178823418263557, 12.50468959796832512801882494495, 13.61875722828907539456556723254