L(s) = 1 | + (−0.840 − 1.45i)2-s + (−1.62 − 2.52i)3-s + (0.585 − 1.01i)4-s + 2.34i·5-s + (−2.31 + 4.48i)6-s + (−3.93 − 5.78i)7-s − 8.69·8-s + (−3.73 + 8.18i)9-s + (3.41 − 1.97i)10-s − 6.20·11-s + (−3.50 + 0.167i)12-s + (21.3 − 12.3i)13-s + (−5.12 + 10.6i)14-s + (5.91 − 3.80i)15-s + (4.97 + 8.61i)16-s + (19.5 − 11.2i)17-s + ⋯ |
L(s) = 1 | + (−0.420 − 0.728i)2-s + (−0.540 − 0.841i)3-s + (0.146 − 0.253i)4-s + 0.468i·5-s + (−0.385 + 0.747i)6-s + (−0.562 − 0.827i)7-s − 1.08·8-s + (−0.415 + 0.909i)9-s + (0.341 − 0.197i)10-s − 0.564·11-s + (−0.292 + 0.0139i)12-s + (1.64 − 0.948i)13-s + (−0.366 + 0.757i)14-s + (0.394 − 0.253i)15-s + (0.310 + 0.538i)16-s + (1.14 − 0.662i)17-s + ⋯ |
Λ(s)=(=(63s/2ΓC(s)L(s)(−0.924+0.381i)Λ(3−s)
Λ(s)=(=(63s/2ΓC(s+1)L(s)(−0.924+0.381i)Λ(1−s)
Degree: |
2 |
Conductor: |
63
= 32⋅7
|
Sign: |
−0.924+0.381i
|
Analytic conductor: |
1.71662 |
Root analytic conductor: |
1.31020 |
Motivic weight: |
2 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ63(61,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 63, ( :1), −0.924+0.381i)
|
Particular Values
L(23) |
≈ |
0.143761−0.725975i |
L(21) |
≈ |
0.143761−0.725975i |
L(2) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1+(1.62+2.52i)T |
| 7 | 1+(3.93+5.78i)T |
good | 2 | 1+(0.840+1.45i)T+(−2+3.46i)T2 |
| 5 | 1−2.34iT−25T2 |
| 11 | 1+6.20T+121T2 |
| 13 | 1+(−21.3+12.3i)T+(84.5−146.i)T2 |
| 17 | 1+(−19.5+11.2i)T+(144.5−250.i)T2 |
| 19 | 1+(9.34+5.39i)T+(180.5+312.i)T2 |
| 23 | 1+8.54T+529T2 |
| 29 | 1+(−16.1+27.9i)T+(−420.5−728.i)T2 |
| 31 | 1+(−1.44−0.833i)T+(480.5+832.i)T2 |
| 37 | 1+(19.7−34.2i)T+(−684.5−1.18e3i)T2 |
| 41 | 1+(27.9−16.1i)T+(840.5−1.45e3i)T2 |
| 43 | 1+(2.15−3.73i)T+(−924.5−1.60e3i)T2 |
| 47 | 1+(−42.0+24.3i)T+(1.10e3−1.91e3i)T2 |
| 53 | 1+(1.04+1.81i)T+(−1.40e3+2.43e3i)T2 |
| 59 | 1+(−91.7−52.9i)T+(1.74e3+3.01e3i)T2 |
| 61 | 1+(20.0−11.5i)T+(1.86e3−3.22e3i)T2 |
| 67 | 1+(−1.29+2.24i)T+(−2.24e3−3.88e3i)T2 |
| 71 | 1+66.6T+5.04e3T2 |
| 73 | 1+(−18.4+10.6i)T+(2.66e3−4.61e3i)T2 |
| 79 | 1+(−51.5−89.3i)T+(−3.12e3+5.40e3i)T2 |
| 83 | 1+(10.0+5.80i)T+(3.44e3+5.96e3i)T2 |
| 89 | 1+(12.0+6.93i)T+(3.96e3+6.85e3i)T2 |
| 97 | 1+(13.0+7.54i)T+(4.70e3+8.14e3i)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
−13.89354279539675408914156458267, −13.00705583344768856354481072879, −11.78830616867558922085472967600, −10.71538796807336084003395189244, −10.17870658943654513413621155559, −8.272730578196727304912230522820, −6.82900992215145405865536078400, −5.75595847309413337872808989244, −3.01938336937109282534644160033, −0.868770893512026453407523140405,
3.57301745582915755406176451562, 5.57312106402715644391871202135, 6.50924974475928480398944676426, 8.445225417174227277147589367089, 9.096704151987272801233137407012, 10.55189286163180638149902956914, 11.90019204968747745860252522727, 12.72605783038464755236279205136, 14.54527374026805120486063797275, 15.75816931332958893703032485421