L(s) = 1 | + (−1.5 + 2.59i)3-s + (−0.5 − 2.59i)7-s + (−3 − 5.19i)9-s + (1 − 1.73i)11-s + 6·13-s + (1 − 1.73i)17-s + (7.5 + 2.59i)21-s + (−4.5 − 7.79i)23-s + 9·27-s + 3·29-s + (−1 + 1.73i)31-s + (3 + 5.19i)33-s + (4 + 6.92i)37-s + (−9 + 15.5i)39-s + 5·41-s + ⋯ |
L(s) = 1 | + (−0.866 + 1.49i)3-s + (−0.188 − 0.981i)7-s + (−1 − 1.73i)9-s + (0.301 − 0.522i)11-s + 1.66·13-s + (0.242 − 0.420i)17-s + (1.63 + 0.566i)21-s + (−0.938 − 1.62i)23-s + 1.73·27-s + 0.557·29-s + (−0.179 + 0.311i)31-s + (0.522 + 0.904i)33-s + (0.657 + 1.13i)37-s + (−1.44 + 2.49i)39-s + 0.780·41-s + ⋯ |
Λ(s)=(=(700s/2ΓC(s)L(s)(0.991−0.126i)Λ(2−s)
Λ(s)=(=(700s/2ΓC(s+1/2)L(s)(0.991−0.126i)Λ(1−s)
Degree: |
2 |
Conductor: |
700
= 22⋅52⋅7
|
Sign: |
0.991−0.126i
|
Analytic conductor: |
5.58952 |
Root analytic conductor: |
2.36421 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ700(501,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 700, ( :1/2), 0.991−0.126i)
|
Particular Values
L(1) |
≈ |
1.07337+0.0681167i |
L(21) |
≈ |
1.07337+0.0681167i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
| 7 | 1+(0.5+2.59i)T |
good | 3 | 1+(1.5−2.59i)T+(−1.5−2.59i)T2 |
| 11 | 1+(−1+1.73i)T+(−5.5−9.52i)T2 |
| 13 | 1−6T+13T2 |
| 17 | 1+(−1+1.73i)T+(−8.5−14.7i)T2 |
| 19 | 1+(−9.5+16.4i)T2 |
| 23 | 1+(4.5+7.79i)T+(−11.5+19.9i)T2 |
| 29 | 1−3T+29T2 |
| 31 | 1+(1−1.73i)T+(−15.5−26.8i)T2 |
| 37 | 1+(−4−6.92i)T+(−18.5+32.0i)T2 |
| 41 | 1−5T+41T2 |
| 43 | 1+T+43T2 |
| 47 | 1+(−4−6.92i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−2+3.46i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−4+6.92i)T+(−29.5−51.0i)T2 |
| 61 | 1+(3.5+6.06i)T+(−30.5+52.8i)T2 |
| 67 | 1+(1.5−2.59i)T+(−33.5−58.0i)T2 |
| 71 | 1−8T+71T2 |
| 73 | 1+(−7+12.1i)T+(−36.5−63.2i)T2 |
| 79 | 1+(2+3.46i)T+(−39.5+68.4i)T2 |
| 83 | 1−T+83T2 |
| 89 | 1+(6.5+11.2i)T+(−44.5+77.0i)T2 |
| 97 | 1−10T+97T2 |
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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
−10.54802795130702890349441450560, −9.862395671494668382720977579197, −8.945986645573651820304592625124, −8.063316455681579894919765870180, −6.48513338097050374821343449940, −6.06971771051922446566618561723, −4.80540911759818867226455012828, −4.06500472574838242289754385963, −3.29358468842960578616381354387, −0.75743476891547149038775290396,
1.23548129594604909976814531111, 2.26010940022169707386274187625, 3.86366605373192957546620990107, 5.60967500654854610223777264096, 5.88680034764386969265989558344, 6.79159175558741715181563945776, 7.72564588396874043121645627425, 8.501938853220002993874118682912, 9.486373957961796894308117388180, 10.73933745385851493502417177774