L(s) = 1 | + (0.0330 + 0.314i)2-s + (1.85 − 0.395i)4-s + (−0.423 + 4.03i)5-s + (2.28 + 2.53i)7-s + (0.380 + 1.17i)8-s − 1.28·10-s + (2.11 + 2.55i)11-s + (1.26 + 0.562i)13-s + (−0.721 + 0.801i)14-s + (3.11 − 1.38i)16-s + (−5.20 − 3.78i)17-s + (−1.07 − 3.31i)19-s + (0.805 + 7.66i)20-s + (−0.733 + 0.749i)22-s + (−1.86 − 3.23i)23-s + ⋯ |
L(s) = 1 | + (0.0233 + 0.222i)2-s + (0.929 − 0.197i)4-s + (−0.189 + 1.80i)5-s + (0.862 + 0.957i)7-s + (0.134 + 0.414i)8-s − 0.405·10-s + (0.637 + 0.770i)11-s + (0.350 + 0.155i)13-s + (−0.192 + 0.214i)14-s + (0.778 − 0.346i)16-s + (−1.26 − 0.916i)17-s + (−0.247 − 0.760i)19-s + (0.180 + 1.71i)20-s + (−0.156 + 0.159i)22-s + (−0.389 − 0.675i)23-s + ⋯ |
Λ(s)=(=(891s/2ΓC(s)L(s)(−0.177−0.984i)Λ(2−s)
Λ(s)=(=(891s/2ΓC(s+1/2)L(s)(−0.177−0.984i)Λ(1−s)
Degree: |
2 |
Conductor: |
891
= 34⋅11
|
Sign: |
−0.177−0.984i
|
Analytic conductor: |
7.11467 |
Root analytic conductor: |
2.66733 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ891(784,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 891, ( :1/2), −0.177−0.984i)
|
Particular Values
L(1) |
≈ |
1.35136+1.61759i |
L(21) |
≈ |
1.35136+1.61759i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 11 | 1+(−2.11−2.55i)T |
good | 2 | 1+(−0.0330−0.314i)T+(−1.95+0.415i)T2 |
| 5 | 1+(0.423−4.03i)T+(−4.89−1.03i)T2 |
| 7 | 1+(−2.28−2.53i)T+(−0.731+6.96i)T2 |
| 13 | 1+(−1.26−0.562i)T+(8.69+9.66i)T2 |
| 17 | 1+(5.20+3.78i)T+(5.25+16.1i)T2 |
| 19 | 1+(1.07+3.31i)T+(−15.3+11.1i)T2 |
| 23 | 1+(1.86+3.23i)T+(−11.5+19.9i)T2 |
| 29 | 1+(0.0499+0.0554i)T+(−3.03+28.8i)T2 |
| 31 | 1+(3.41+1.52i)T+(20.7+23.0i)T2 |
| 37 | 1+(0.381−1.17i)T+(−29.9−21.7i)T2 |
| 41 | 1+(−6.23+6.92i)T+(−4.28−40.7i)T2 |
| 43 | 1+(−0.228+0.395i)T+(−21.5−37.2i)T2 |
| 47 | 1+(−2.14−0.456i)T+(42.9+19.1i)T2 |
| 53 | 1+(−0.729+0.530i)T+(16.3−50.4i)T2 |
| 59 | 1+(−7.84+1.66i)T+(53.8−23.9i)T2 |
| 61 | 1+(−5.27+2.34i)T+(40.8−45.3i)T2 |
| 67 | 1+(−2.92−5.06i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−7.66−5.56i)T+(21.9+67.5i)T2 |
| 73 | 1+(−4.34+13.3i)T+(−59.0−42.9i)T2 |
| 79 | 1+(−1.14−10.9i)T+(−77.2+16.4i)T2 |
| 83 | 1+(1.24−0.556i)T+(55.5−61.6i)T2 |
| 89 | 1+10.3T+89T2 |
| 97 | 1+(−0.862−8.20i)T+(−94.8+20.1i)T2 |
show more | |
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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
−10.63918531720483636255908670847, −9.600737831240384404485081963836, −8.590286442268777293937036322865, −7.50974469257793913183993660902, −6.85334176598138516949563543144, −6.37460816742549666309175196742, −5.26620668212842863655888525404, −3.97709971939137004679562882907, −2.47885661618552087298043627925, −2.21058739083045129476485213465,
1.05973239339167738258236659489, 1.81713596381233582483468384259, 3.80048538623248192805025457151, 4.24416071049026031512706242629, 5.49225556717972759420145268425, 6.37213039182549431993631535542, 7.57327473417837125750663133940, 8.253031642245758603927057083607, 8.823649820796953535378813767867, 9.956578039245776300568890379081