L(s) = 1 | + (1.12 + 1.94i)2-s + (0.277 + 0.480i)3-s + (−1.52 + 2.64i)4-s − 1.44·5-s + (−0.623 + 1.07i)6-s + (−1.02 + 1.77i)7-s − 2.35·8-s + (1.34 − 2.33i)9-s + (−1.62 − 2.81i)10-s + (1.27 + 2.21i)11-s − 1.69·12-s − 4.60·14-s + (−0.400 − 0.694i)15-s + (0.400 + 0.694i)16-s + (2.64 − 4.58i)17-s + 6.04·18-s + ⋯ |
L(s) = 1 | + (0.794 + 1.37i)2-s + (0.160 + 0.277i)3-s + (−0.762 + 1.32i)4-s − 0.646·5-s + (−0.254 + 0.440i)6-s + (−0.387 + 0.670i)7-s − 0.833·8-s + (0.448 − 0.777i)9-s + (−0.513 − 0.889i)10-s + (0.385 + 0.667i)11-s − 0.488·12-s − 1.23·14-s + (−0.103 − 0.179i)15-s + (0.100 + 0.173i)16-s + (0.642 − 1.11i)17-s + 1.42·18-s + ⋯ |
Λ(s)=(=(169s/2ΓC(s)L(s)(−0.611−0.791i)Λ(2−s)
Λ(s)=(=(169s/2ΓC(s+1/2)L(s)(−0.611−0.791i)Λ(1−s)
Degree: |
2 |
Conductor: |
169
= 132
|
Sign: |
−0.611−0.791i
|
Analytic conductor: |
1.34947 |
Root analytic conductor: |
1.16166 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ169(146,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 169, ( :1/2), −0.611−0.791i)
|
Particular Values
L(1) |
≈ |
0.702135+1.43030i |
L(21) |
≈ |
0.702135+1.43030i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 13 | 1 |
good | 2 | 1+(−1.12−1.94i)T+(−1+1.73i)T2 |
| 3 | 1+(−0.277−0.480i)T+(−1.5+2.59i)T2 |
| 5 | 1+1.44T+5T2 |
| 7 | 1+(1.02−1.77i)T+(−3.5−6.06i)T2 |
| 11 | 1+(−1.27−2.21i)T+(−5.5+9.52i)T2 |
| 17 | 1+(−2.64+4.58i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−2.92+5.06i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−0.945−1.63i)T+(−11.5+19.9i)T2 |
| 29 | 1+(1.13+1.96i)T+(−14.5+25.1i)T2 |
| 31 | 1+4.26T+31T2 |
| 37 | 1+(2.67+4.63i)T+(−18.5+32.0i)T2 |
| 41 | 1+(0.637+1.10i)T+(−20.5+35.5i)T2 |
| 43 | 1+(3.06−5.31i)T+(−21.5−37.2i)T2 |
| 47 | 1+2.95T+47T2 |
| 53 | 1−5.52T+53T2 |
| 59 | 1+(−6.10+10.5i)T+(−29.5−51.0i)T2 |
| 61 | 1+(4.28−7.41i)T+(−30.5−52.8i)T2 |
| 67 | 1+(0.288+0.499i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−2.29+3.97i)T+(−35.5−61.4i)T2 |
| 73 | 1+10.5T+73T2 |
| 79 | 1+15.7T+79T2 |
| 83 | 1−7.72T+83T2 |
| 89 | 1+(3.30+5.72i)T+(−44.5+77.0i)T2 |
| 97 | 1+(5.96−10.3i)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
−13.31293592486676508382710946243, −12.37052849792068554718385194965, −11.56442217790137039574434957359, −9.735891870165948494256853100657, −8.939379714636231942686446275645, −7.51668844787155801207694639329, −6.90642977775259276233257201906, −5.63052577988177386537391055901, −4.51962889787279237510927532994, −3.37132537688323264529287661625,
1.52689883213530529365582765022, 3.37016327596816426240421462671, 4.10040853589825648168981381951, 5.58180037649658538309998755809, 7.25731000500646181093224426441, 8.318256106869341408267950777152, 10.00325808879200294697827716316, 10.57893161583639180765487849826, 11.61041980441695972904060873771, 12.40906186397483256211144051925