L(s) = 1 | + (−1.73 + i)2-s + (1.47 + 2.56i)3-s + (1.99 − 3.46i)4-s + 20.2i·5-s + (−5.12 − 2.95i)6-s + (−1.04 − 0.603i)7-s + 7.99i·8-s + (9.13 − 15.8i)9-s + (−20.2 − 35.0i)10-s + (35.3 − 20.4i)11-s + 11.8·12-s + (−36.8 − 29.0i)13-s + 2.41·14-s + (−51.7 + 29.9i)15-s + (−8 − 13.8i)16-s + (4.90 − 8.49i)17-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.284 + 0.492i)3-s + (0.249 − 0.433i)4-s + 1.80i·5-s + (−0.348 − 0.201i)6-s + (−0.0564 − 0.0326i)7-s + 0.353i·8-s + (0.338 − 0.585i)9-s + (−0.639 − 1.10i)10-s + (0.969 − 0.559i)11-s + 0.284·12-s + (−0.785 − 0.618i)13-s + 0.0461·14-s + (−0.891 + 0.514i)15-s + (−0.125 − 0.216i)16-s + (0.0699 − 0.121i)17-s + ⋯ |
Λ(s)=(=(26s/2ΓC(s)L(s)(0.155−0.987i)Λ(4−s)
Λ(s)=(=(26s/2ΓC(s+3/2)L(s)(0.155−0.987i)Λ(1−s)
Degree: |
2 |
Conductor: |
26
= 2⋅13
|
Sign: |
0.155−0.987i
|
Analytic conductor: |
1.53404 |
Root analytic conductor: |
1.23856 |
Motivic weight: |
3 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ26(23,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 26, ( :3/2), 0.155−0.987i)
|
Particular Values
L(2) |
≈ |
0.740469+0.632717i |
L(21) |
≈ |
0.740469+0.632717i |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(1.73−i)T |
| 13 | 1+(36.8+29.0i)T |
good | 3 | 1+(−1.47−2.56i)T+(−13.5+23.3i)T2 |
| 5 | 1−20.2iT−125T2 |
| 7 | 1+(1.04+0.603i)T+(171.5+297.i)T2 |
| 11 | 1+(−35.3+20.4i)T+(665.5−1.15e3i)T2 |
| 17 | 1+(−4.90+8.49i)T+(−2.45e3−4.25e3i)T2 |
| 19 | 1+(−98.7−57.0i)T+(3.42e3+5.94e3i)T2 |
| 23 | 1+(11.6+20.1i)T+(−6.08e3+1.05e4i)T2 |
| 29 | 1+(−31.7−55.0i)T+(−1.21e4+2.11e4i)T2 |
| 31 | 1+225.iT−2.97e4T2 |
| 37 | 1+(154.−89.2i)T+(2.53e4−4.38e4i)T2 |
| 41 | 1+(70.6−40.7i)T+(3.44e4−5.96e4i)T2 |
| 43 | 1+(−164.+284.i)T+(−3.97e4−6.88e4i)T2 |
| 47 | 1−560.iT−1.03e5T2 |
| 53 | 1+389.T+1.48e5T2 |
| 59 | 1+(189.+109.i)T+(1.02e5+1.77e5i)T2 |
| 61 | 1+(−180.+311.i)T+(−1.13e5−1.96e5i)T2 |
| 67 | 1+(475.−274.i)T+(1.50e5−2.60e5i)T2 |
| 71 | 1+(689.+397.i)T+(1.78e5+3.09e5i)T2 |
| 73 | 1−140.iT−3.89e5T2 |
| 79 | 1−568.T+4.93e5T2 |
| 83 | 1+1.09e3iT−5.71e5T2 |
| 89 | 1+(760.−438.i)T+(3.52e5−6.10e5i)T2 |
| 97 | 1+(−1.37e3−794.i)T+(4.56e5+7.90e5i)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
−17.40827400669621164589412251854, −15.85968883597927490997689957648, −14.81576164919266349511202079888, −14.17864357073559192369067309043, −11.75824269503622841056602178866, −10.40965042958356472520669941294, −9.498325387408235420420355844433, −7.53584774224580991021563361460, −6.29622170845039603083225160867, −3.30659998857122552043519384155,
1.49915896595163907660851419763, 4.73283295372721493181530228728, 7.32897958601201697832511984849, 8.735416058314895116692319527460, 9.694689234930773539898081948233, 11.85844185208383344165663643858, 12.64761183126550726043814731781, 13.88148675608352281793243595311, 15.92486906392024615489433126240, 16.82869838399006707348411536987