L(s) = 1 | + (−0.288 − 1.38i)2-s + (0.450 − 0.780i)3-s + (−1.83 + 0.798i)4-s + (−1.21 − 0.398i)6-s + (2.29 − 1.30i)7-s + (1.63 + 2.30i)8-s + (1.09 + 1.89i)9-s + (3.24 + 1.87i)11-s + (−0.202 + 1.79i)12-s + 2.41i·13-s + (−2.47 − 2.80i)14-s + (2.72 − 2.92i)16-s + (0.505 + 0.291i)17-s + (2.30 − 2.06i)18-s + (3.07 + 5.33i)19-s + ⋯ |
L(s) = 1 | + (−0.204 − 0.978i)2-s + (0.260 − 0.450i)3-s + (−0.916 + 0.399i)4-s + (−0.494 − 0.162i)6-s + (0.869 − 0.494i)7-s + (0.578 + 0.815i)8-s + (0.364 + 0.631i)9-s + (0.977 + 0.564i)11-s + (−0.0585 + 0.517i)12-s + 0.671i·13-s + (−0.661 − 0.750i)14-s + (0.680 − 0.732i)16-s + (0.122 + 0.0707i)17-s + (0.543 − 0.485i)18-s + (0.706 + 1.22i)19-s + ⋯ |
Λ(s)=(=(700s/2ΓC(s)L(s)(0.506+0.862i)Λ(2−s)
Λ(s)=(=(700s/2ΓC(s+1/2)L(s)(0.506+0.862i)Λ(1−s)
Degree: |
2 |
Conductor: |
700
= 22⋅52⋅7
|
Sign: |
0.506+0.862i
|
Analytic conductor: |
5.58952 |
Root analytic conductor: |
2.36421 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ700(451,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 700, ( :1/2), 0.506+0.862i)
|
Particular Values
L(1) |
≈ |
1.40383−0.803268i |
L(21) |
≈ |
1.40383−0.803268i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1+(0.288+1.38i)T |
| 5 | 1 |
| 7 | 1+(−2.29+1.30i)T |
good | 3 | 1+(−0.450+0.780i)T+(−1.5−2.59i)T2 |
| 11 | 1+(−3.24−1.87i)T+(5.5+9.52i)T2 |
| 13 | 1−2.41iT−13T2 |
| 17 | 1+(−0.505−0.291i)T+(8.5+14.7i)T2 |
| 19 | 1+(−3.07−5.33i)T+(−9.5+16.4i)T2 |
| 23 | 1+(3.73−2.15i)T+(11.5−19.9i)T2 |
| 29 | 1+0.435T+29T2 |
| 31 | 1+(−1.26+2.19i)T+(−15.5−26.8i)T2 |
| 37 | 1+(5.65+9.78i)T+(−18.5+32.0i)T2 |
| 41 | 1+7.35iT−41T2 |
| 43 | 1+5.80iT−43T2 |
| 47 | 1+(−5.78−10.0i)T+(−23.5+40.7i)T2 |
| 53 | 1+(1.55−2.69i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−1.73+3.00i)T+(−29.5−51.0i)T2 |
| 61 | 1+(8.99−5.19i)T+(30.5−52.8i)T2 |
| 67 | 1+(−8.52−4.92i)T+(33.5+58.0i)T2 |
| 71 | 1+9.96iT−71T2 |
| 73 | 1+(8.48+4.89i)T+(36.5+63.2i)T2 |
| 79 | 1+(−0.397+0.229i)T+(39.5−68.4i)T2 |
| 83 | 1+2.59T+83T2 |
| 89 | 1+(8.55−4.94i)T+(44.5−77.0i)T2 |
| 97 | 1−4.54iT−97T2 |
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
−10.39295363427268366571405206908, −9.552315658691274286267401492955, −8.685078148088543470546396724485, −7.70458216584139071539264597139, −7.24505093422419116895140097668, −5.61994972933546055765557827627, −4.41969111333366381119231629663, −3.78140005486574857981845541544, −2.06334771878622122453001655337, −1.42747652266068465734193393270,
1.13199219369193030973908726704, 3.21440342933127530425380718191, 4.37439883188040736035252330151, 5.18473240947939256646899729109, 6.23258650267539193696138988422, 7.03984878064138429624432859154, 8.193359014806216574932042231883, 8.724924930497671790525700281345, 9.517320483380932721221831261726, 10.26868032315461018657650697784