L(s) = 1 | − 0.579·2-s + (0.946 + 1.63i)3-s − 1.66·4-s + (−0.736 − 1.27i)5-s + (−0.548 − 0.950i)6-s + 2.12·8-s + (−0.289 + 0.502i)9-s + (0.427 + 0.739i)10-s + (−0.289 − 0.502i)11-s + (−1.57 − 2.72i)12-s + (−0.128 − 3.60i)13-s + (1.39 − 2.41i)15-s + 2.09·16-s − 1.19·17-s + (0.168 − 0.291i)18-s + (0.230 − 0.399i)19-s + ⋯ |
L(s) = 1 | − 0.409·2-s + (0.546 + 0.946i)3-s − 0.831·4-s + (−0.329 − 0.570i)5-s + (−0.223 − 0.387i)6-s + 0.751·8-s + (−0.0966 + 0.167i)9-s + (0.135 + 0.233i)10-s + (−0.0874 − 0.151i)11-s + (−0.454 − 0.787i)12-s + (−0.0357 − 0.999i)13-s + (0.359 − 0.623i)15-s + 0.523·16-s − 0.290·17-s + (0.0396 − 0.0686i)18-s + (0.0528 − 0.0915i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.908+0.418i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.908+0.418i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.908+0.418i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(165,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.908+0.418i)
|
Particular Values
L(1) |
≈ |
1.03007−0.225931i |
L(21) |
≈ |
1.03007−0.225931i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(0.128+3.60i)T |
good | 2 | 1+0.579T+2T2 |
| 3 | 1+(−0.946−1.63i)T+(−1.5+2.59i)T2 |
| 5 | 1+(0.736+1.27i)T+(−2.5+4.33i)T2 |
| 11 | 1+(0.289+0.502i)T+(−5.5+9.52i)T2 |
| 17 | 1+1.19T+17T2 |
| 19 | 1+(−0.230+0.399i)T+(−9.5−16.4i)T2 |
| 23 | 1−2.36T+23T2 |
| 29 | 1+(−3.44+5.96i)T+(−14.5−25.1i)T2 |
| 31 | 1+(2.22−3.84i)T+(−15.5−26.8i)T2 |
| 37 | 1−9.16T+37T2 |
| 41 | 1+(−2.00+3.47i)T+(−20.5−35.5i)T2 |
| 43 | 1+(4.02+6.97i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−5.75−9.97i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−4.69+8.13i)T+(−26.5−45.8i)T2 |
| 59 | 1−0.240T+59T2 |
| 61 | 1+(3.86−6.69i)T+(−30.5−52.8i)T2 |
| 67 | 1+(−0.724−1.25i)T+(−33.5+58.0i)T2 |
| 71 | 1+(−6.25−10.8i)T+(−35.5+61.4i)T2 |
| 73 | 1+(−1.84+3.19i)T+(−36.5−63.2i)T2 |
| 79 | 1+(8.03+13.9i)T+(−39.5+68.4i)T2 |
| 83 | 1+15.4T+83T2 |
| 89 | 1−2.49T+89T2 |
| 97 | 1+(7.82+13.5i)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
−10.18959329775372536320392787884, −9.654977590258728194680102112850, −8.676126221932964059805741573621, −8.409329335751048881566816964427, −7.30773792968958589999772821328, −5.73083036052229599421266503381, −4.68330006120752527351014588015, −4.09840852892491596434006468933, −2.93763734508302235611660294863, −0.74268881960285116168918604274,
1.32204334879384365880013536142, 2.67339105952063251936383461332, 3.99895295430819206552498111960, 5.03406990273487663456892356921, 6.54306378088053582098632234821, 7.29700966037458428552247362415, 7.977737751710605754359507400566, 8.843305502601304640024906774023, 9.542638295138275149120493460150, 10.60137771255938385272767012646