L(s) = 1 | + 3.22i·5-s + (0.866 + 0.5i)7-s + (−1.17 + 0.678i)11-s + (1.95 − 3.03i)13-s + (−1.44 + 2.50i)17-s + (5.05 + 2.92i)19-s + (0.482 + 0.835i)23-s − 5.39·25-s + (3.61 + 6.26i)29-s − 2.60i·31-s + (−1.61 + 2.79i)35-s + (−4.35 + 2.51i)37-s + (8.03 − 4.63i)41-s + (0.226 − 0.392i)43-s + 5.42i·47-s + ⋯ |
L(s) = 1 | + 1.44i·5-s + (0.327 + 0.188i)7-s + (−0.354 + 0.204i)11-s + (0.541 − 0.840i)13-s + (−0.351 + 0.608i)17-s + (1.16 + 0.669i)19-s + (0.100 + 0.174i)23-s − 1.07·25-s + (0.672 + 1.16i)29-s − 0.468i·31-s + (−0.272 + 0.471i)35-s + (−0.715 + 0.413i)37-s + (1.25 − 0.724i)41-s + (0.0345 − 0.0597i)43-s + 0.791i·47-s + ⋯ |
Λ(s)=(=(3276s/2ΓC(s)L(s)(−0.446−0.894i)Λ(2−s)
Λ(s)=(=(3276s/2ΓC(s+1/2)L(s)(−0.446−0.894i)Λ(1−s)
Degree: |
2 |
Conductor: |
3276
= 22⋅32⋅7⋅13
|
Sign: |
−0.446−0.894i
|
Analytic conductor: |
26.1589 |
Root analytic conductor: |
5.11458 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ3276(1765,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 3276, ( :1/2), −0.446−0.894i)
|
Particular Values
L(1) |
≈ |
1.736743986 |
L(21) |
≈ |
1.736743986 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 7 | 1+(−0.866−0.5i)T |
| 13 | 1+(−1.95+3.03i)T |
good | 5 | 1−3.22iT−5T2 |
| 11 | 1+(1.17−0.678i)T+(5.5−9.52i)T2 |
| 17 | 1+(1.44−2.50i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−5.05−2.92i)T+(9.5+16.4i)T2 |
| 23 | 1+(−0.482−0.835i)T+(−11.5+19.9i)T2 |
| 29 | 1+(−3.61−6.26i)T+(−14.5+25.1i)T2 |
| 31 | 1+2.60iT−31T2 |
| 37 | 1+(4.35−2.51i)T+(18.5−32.0i)T2 |
| 41 | 1+(−8.03+4.63i)T+(20.5−35.5i)T2 |
| 43 | 1+(−0.226+0.392i)T+(−21.5−37.2i)T2 |
| 47 | 1−5.42iT−47T2 |
| 53 | 1+1.37T+53T2 |
| 59 | 1+(2.69+1.55i)T+(29.5+51.0i)T2 |
| 61 | 1+(2.31−4.01i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−0.232+0.134i)T+(33.5−58.0i)T2 |
| 71 | 1+(−10.6−6.13i)T+(35.5+61.4i)T2 |
| 73 | 1−9.24iT−73T2 |
| 79 | 1+8.13T+79T2 |
| 83 | 1−2.59iT−83T2 |
| 89 | 1+(5.84−3.37i)T+(44.5−77.0i)T2 |
| 97 | 1+(11.5+6.67i)T+(48.5+84.0i)T2 |
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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
−8.786900208582508526803347153905, −8.007283441063841712041877182777, −7.40456663185525336326357133723, −6.69996396994302279398049709297, −5.87245814861654236388179837074, −5.26970089693466565659212351543, −4.05003670646927070630669932925, −3.21422364286908981294099760357, −2.58458015539271588440802076633, −1.35191363878764940238097246791,
0.57025402115164352141845504660, 1.48018979723630319712962674001, 2.65840288494546494026273136037, 3.86678737375396869718616462893, 4.70641949597410112116939737256, 5.11671415657919898394414771198, 6.05867835262213709934843414493, 6.96246519023806239196627076703, 7.81854606601088230421911667684, 8.438632177714979027689648394236