L(s) = 1 | + (−1.26 − 1.26i)2-s − 1.73·3-s − 0.798i·4-s + (−4.65 − 4.65i)5-s + (2.19 + 2.19i)6-s + (1.87 − 1.87i)7-s + (−6.07 + 6.07i)8-s + 2.99·9-s + 11.7i·10-s + (11.0 − 11.0i)11-s + 1.38i·12-s + (11.4 + 6.06i)13-s − 4.75·14-s + (8.06 + 8.06i)15-s + 12.1·16-s + 3.20i·17-s + ⋯ |
L(s) = 1 | + (−0.632 − 0.632i)2-s − 0.577·3-s − 0.199i·4-s + (−0.931 − 0.931i)5-s + (0.365 + 0.365i)6-s + (0.268 − 0.268i)7-s + (−0.758 + 0.758i)8-s + 0.333·9-s + 1.17i·10-s + (1.00 − 1.00i)11-s + 0.115i·12-s + (0.884 + 0.466i)13-s − 0.339·14-s + (0.537 + 0.537i)15-s + 0.760·16-s + 0.188i·17-s + ⋯ |
Λ(s)=(=(39s/2ΓC(s)L(s)(−0.702+0.711i)Λ(3−s)
Λ(s)=(=(39s/2ΓC(s+1)L(s)(−0.702+0.711i)Λ(1−s)
Degree: |
2 |
Conductor: |
39
= 3⋅13
|
Sign: |
−0.702+0.711i
|
Analytic conductor: |
1.06267 |
Root analytic conductor: |
1.03086 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ39(34,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 39, ( :1), −0.702+0.711i)
|
Particular Values
L(23) |
≈ |
0.212229−0.508118i |
L(21) |
≈ |
0.212229−0.508118i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+1.73T |
| 13 | 1+(−11.4−6.06i)T |
good | 2 | 1+(1.26+1.26i)T+4iT2 |
| 5 | 1+(4.65+4.65i)T+25iT2 |
| 7 | 1+(−1.87+1.87i)T−49iT2 |
| 11 | 1+(−11.0+11.0i)T−121iT2 |
| 17 | 1−3.20iT−289T2 |
| 19 | 1+(19.7+19.7i)T+361iT2 |
| 23 | 1−3.84iT−529T2 |
| 29 | 1−25.4T+841T2 |
| 31 | 1+(25.7+25.7i)T+961iT2 |
| 37 | 1+(−35.2+35.2i)T−1.36e3iT2 |
| 41 | 1+(−41.0−41.0i)T+1.68e3iT2 |
| 43 | 1−51.1iT−1.84e3T2 |
| 47 | 1+(−0.856+0.856i)T−2.20e3iT2 |
| 53 | 1+39.2T+2.80e3T2 |
| 59 | 1+(−54.7+54.7i)T−3.48e3iT2 |
| 61 | 1−50.5T+3.72e3T2 |
| 67 | 1+(1.32+1.32i)T+4.48e3iT2 |
| 71 | 1+(10.8+10.8i)T+5.04e3iT2 |
| 73 | 1+(91.2−91.2i)T−5.32e3iT2 |
| 79 | 1−95.2T+6.24e3T2 |
| 83 | 1+(20.5+20.5i)T+6.88e3iT2 |
| 89 | 1+(71.2−71.2i)T−7.92e3iT2 |
| 97 | 1+(−28.4−28.4i)T+9.40e3iT2 |
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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
−15.90229601459738281437847430901, −14.46300071822782809327746035646, −12.85100623723419785205862558093, −11.39456648775723331396706858319, −11.14493292080846773750196858265, −9.268253990503975919829497548866, −8.319725717180099591951039635340, −6.16912531339483720909952502734, −4.30207349851120949249536880178, −0.893582822051883630884892976829,
3.86473272019059703044966353948, 6.37323303951126276055559704336, 7.38752350060111876012239379157, 8.647719334869752701665622913581, 10.34146852372127981291283636087, 11.65945856964439467475927562875, 12.55468169466272420335860685679, 14.65231727225159388066840835163, 15.45177608089529926143625556347, 16.48337310709179536544459443650