L(s) = 1 | + 2.08·2-s + (1.31 + 1.31i)3-s + 2.34·4-s + (2.73 + 2.73i)6-s + (0.971 − 0.971i)7-s + 0.713·8-s + 0.435i·9-s + (−0.384 − 0.384i)11-s + (3.07 + 3.07i)12-s + 5.39i·13-s + (2.02 − 2.02i)14-s − 3.19·16-s + (−3.38 − 2.36i)17-s + 0.907i·18-s − 4.86i·19-s + ⋯ |
L(s) = 1 | + 1.47·2-s + (0.756 + 0.756i)3-s + 1.17·4-s + (1.11 + 1.11i)6-s + (0.367 − 0.367i)7-s + 0.252·8-s + 0.145i·9-s + (−0.116 − 0.116i)11-s + (0.886 + 0.886i)12-s + 1.49i·13-s + (0.541 − 0.541i)14-s − 0.799·16-s + (−0.819 − 0.572i)17-s + 0.213i·18-s − 1.11i·19-s + ⋯ |
Λ(s)=(=(425s/2ΓC(s)L(s)(0.869−0.494i)Λ(2−s)
Λ(s)=(=(425s/2ΓC(s+1/2)L(s)(0.869−0.494i)Λ(1−s)
Degree: |
2 |
Conductor: |
425
= 52⋅17
|
Sign: |
0.869−0.494i
|
Analytic conductor: |
3.39364 |
Root analytic conductor: |
1.84218 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ425(174,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 425, ( :1/2), 0.869−0.494i)
|
Particular Values
L(1) |
≈ |
3.40001+0.899242i |
L(21) |
≈ |
3.40001+0.899242i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 17 | 1+(3.38+2.36i)T |
good | 2 | 1−2.08T+2T2 |
| 3 | 1+(−1.31−1.31i)T+3iT2 |
| 7 | 1+(−0.971+0.971i)T−7iT2 |
| 11 | 1+(0.384+0.384i)T+11iT2 |
| 13 | 1−5.39iT−13T2 |
| 19 | 1+4.86iT−19T2 |
| 23 | 1+(−1.26+1.26i)T−23iT2 |
| 29 | 1+(1.29−1.29i)T−29iT2 |
| 31 | 1+(5.73−5.73i)T−31iT2 |
| 37 | 1+(−4.22−4.22i)T+37iT2 |
| 41 | 1+(2.70+2.70i)T+41iT2 |
| 43 | 1−3.66T+43T2 |
| 47 | 1+9.07iT−47T2 |
| 53 | 1−10.5T+53T2 |
| 59 | 1−6.52iT−59T2 |
| 61 | 1+(10.9+10.9i)T+61iT2 |
| 67 | 1−5.68iT−67T2 |
| 71 | 1+(−0.749+0.749i)T−71iT2 |
| 73 | 1+(10.3+10.3i)T+73iT2 |
| 79 | 1+(−0.878−0.878i)T+79iT2 |
| 83 | 1−13.5T+83T2 |
| 89 | 1−0.989T+89T2 |
| 97 | 1+(−8.05−8.05i)T+97iT2 |
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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
−11.43298746781267548964061735943, −10.59836423784515693373421750544, −9.141813569127662224332021373323, −8.925853909970098537672238257347, −7.21127928498020536871309903646, −6.44917140543163931611320772518, −4.96962827011359248733870219314, −4.40306697893599544764131823076, −3.49519872253261796584709807914, −2.36571111800565000415472227211,
2.00676258610101639636567116562, 2.99112591469537480019549339482, 4.12446143213451351825118637143, 5.38489501076848187278286785769, 6.06627068763835988397042766777, 7.39469997429934234590145393688, 8.082833390055605139890840500259, 9.081108984291045723129052624262, 10.49803016406350202512746994576, 11.41123189110714874994576540397