L(s) = 1 | + (0.113 + 0.0656i)2-s + (−0.0890 − 0.332i)3-s + (−0.991 − 1.71i)4-s + (2.08 + 0.813i)5-s + (0.0116 − 0.0436i)6-s + (1.39 + 2.40i)7-s − 0.522i·8-s + (2.49 − 1.44i)9-s + (0.183 + 0.229i)10-s + (1.04 + 3.91i)11-s + (−0.482 + 0.482i)12-s + 0.365i·14-s + (0.0847 − 0.764i)15-s + (−1.94 + 3.37i)16-s + (2.34 + 0.627i)17-s + 0.378·18-s + ⋯ |
L(s) = 1 | + (0.0804 + 0.0464i)2-s + (−0.0513 − 0.191i)3-s + (−0.495 − 0.858i)4-s + (0.931 + 0.363i)5-s + (0.00477 − 0.0178i)6-s + (0.525 + 0.910i)7-s − 0.184i·8-s + (0.831 − 0.480i)9-s + (0.0580 + 0.0724i)10-s + (0.316 + 1.18i)11-s + (−0.139 + 0.139i)12-s + 0.0976i·14-s + (0.0218 − 0.197i)15-s + (−0.487 + 0.843i)16-s + (0.567 + 0.152i)17-s + 0.0891·18-s + ⋯ |
Λ(s)=(=(845s/2ΓC(s)L(s)(0.989−0.142i)Λ(2−s)
Λ(s)=(=(845s/2ΓC(s+1/2)L(s)(0.989−0.142i)Λ(1−s)
Degree: |
2 |
Conductor: |
845
= 5⋅132
|
Sign: |
0.989−0.142i
|
Analytic conductor: |
6.74735 |
Root analytic conductor: |
2.59756 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ845(657,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 845, ( :1/2), 0.989−0.142i)
|
Particular Values
L(1) |
≈ |
1.88541+0.134816i |
L(21) |
≈ |
1.88541+0.134816i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1+(−2.08−0.813i)T |
| 13 | 1 |
good | 2 | 1+(−0.113−0.0656i)T+(1+1.73i)T2 |
| 3 | 1+(0.0890+0.332i)T+(−2.59+1.5i)T2 |
| 7 | 1+(−1.39−2.40i)T+(−3.5+6.06i)T2 |
| 11 | 1+(−1.04−3.91i)T+(−9.52+5.5i)T2 |
| 17 | 1+(−2.34−0.627i)T+(14.7+8.5i)T2 |
| 19 | 1+(1.83+0.491i)T+(16.4+9.5i)T2 |
| 23 | 1+(7.70−2.06i)T+(19.9−11.5i)T2 |
| 29 | 1+(−3.96−2.28i)T+(14.5+25.1i)T2 |
| 31 | 1+(−3.87−3.87i)T+31iT2 |
| 37 | 1+(−3.50+6.07i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−6.20+1.66i)T+(35.5−20.5i)T2 |
| 43 | 1+(−1.67+6.24i)T+(−37.2−21.5i)T2 |
| 47 | 1+0.512T+47T2 |
| 53 | 1+(1.32−1.32i)T−53iT2 |
| 59 | 1+(0.679−2.53i)T+(−51.0−29.5i)T2 |
| 61 | 1+(−0.641−1.11i)T+(−30.5+52.8i)T2 |
| 67 | 1+(3.13+1.80i)T+(33.5+58.0i)T2 |
| 71 | 1+(−1.66+6.20i)T+(−61.4−35.5i)T2 |
| 73 | 1+9.93iT−73T2 |
| 79 | 1+8.37iT−79T2 |
| 83 | 1−3.17T+83T2 |
| 89 | 1+(6.01−1.61i)T+(77.0−44.5i)T2 |
| 97 | 1+(10.1−5.88i)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
−10.07310233859061812701025365811, −9.511421910111506388797782605854, −8.800378168039940777208032780980, −7.53971989753605937851294345060, −6.51503943642436413992641683480, −5.88853349905496892071778732952, −4.99755930379792521766285254722, −4.05967283751629890938257116683, −2.23985670328596479533782074102, −1.45856498846809771401175638751,
1.10444289806714292029363608533, 2.64582985170221863960370322375, 4.08129605018566613761265586288, 4.52170398446535647486764146398, 5.71551157467651672696846289150, 6.69954850673577266854068810681, 8.075813010942802114908552582846, 8.137056593464509927238417286020, 9.490693204031898793603953957498, 10.04508982909926836715196009754