L(s) = 1 | + (1.16 − 2.01i)2-s + (−1.15 + 1.99i)3-s + (−1.71 − 2.97i)4-s + 3.37·5-s + (2.69 + 4.66i)6-s − 3.34·8-s + (−1.16 − 2.01i)9-s + (3.92 − 6.80i)10-s + (−1.16 + 2.01i)11-s + 7.93·12-s + (0.408 − 3.58i)13-s + (−3.89 + 6.74i)15-s + (−0.466 + 0.808i)16-s + (2.72 + 4.72i)17-s − 5.43·18-s + (3.58 + 6.20i)19-s + ⋯ |
L(s) = 1 | + (0.824 − 1.42i)2-s + (−0.666 + 1.15i)3-s + (−0.858 − 1.48i)4-s + 1.50·5-s + (1.09 + 1.90i)6-s − 1.18·8-s + (−0.388 − 0.673i)9-s + (1.24 − 2.15i)10-s + (−0.351 + 0.608i)11-s + 2.29·12-s + (0.113 − 0.993i)13-s + (−1.00 + 1.74i)15-s + (−0.116 + 0.202i)16-s + (0.661 + 1.14i)17-s − 1.28·18-s + (0.822 + 1.42i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.617+0.786i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.617+0.786i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.617+0.786i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(295,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.617+0.786i)
|
Particular Values
L(1) |
≈ |
2.09620−1.01933i |
L(21) |
≈ |
2.09620−1.01933i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(−0.408+3.58i)T |
good | 2 | 1+(−1.16+2.01i)T+(−1−1.73i)T2 |
| 3 | 1+(1.15−1.99i)T+(−1.5−2.59i)T2 |
| 5 | 1−3.37T+5T2 |
| 11 | 1+(1.16−2.01i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−2.72−4.72i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−3.58−6.20i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−3.22+5.58i)T+(−11.5−19.9i)T2 |
| 29 | 1+(−4.22+7.31i)T+(−14.5−25.1i)T2 |
| 31 | 1+3.05T+31T2 |
| 37 | 1+(1.52−2.64i)T+(−18.5−32.0i)T2 |
| 41 | 1+(−0.468+0.812i)T+(−20.5−35.5i)T2 |
| 43 | 1+(−2.04−3.54i)T+(−21.5+37.2i)T2 |
| 47 | 1+3.46T+47T2 |
| 53 | 1+2.34T+53T2 |
| 59 | 1+(3.62+6.27i)T+(−29.5+51.0i)T2 |
| 61 | 1+(3.19+5.53i)T+(−30.5+52.8i)T2 |
| 67 | 1+(2.30−3.99i)T+(−33.5−58.0i)T2 |
| 71 | 1+(−3.79−6.57i)T+(−35.5+61.4i)T2 |
| 73 | 1+2.06T+73T2 |
| 79 | 1+7.58T+79T2 |
| 83 | 1+2.89T+83T2 |
| 89 | 1+(6.57−11.3i)T+(−44.5−77.0i)T2 |
| 97 | 1+(−1.77−3.08i)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
−10.39193177549276948630690102050, −10.02881473203841458105950706452, −9.569227284825654373202355974365, −8.042721997994948108429505932805, −6.12933594392751428547040942354, −5.52466898490866648135221072432, −4.85767353539839806071901066988, −3.82108781491715737917609565054, −2.71465563260493320135431246918, −1.47572498512007748167774136262,
1.39684944977077239307035708651, 3.03371630124749315740872860735, 4.94511416817988465295532594209, 5.49150699438762978378239730757, 6.17443488899378889804159123925, 7.10417673830165012319683845128, 7.29640587639758682251738853503, 8.825592048779301317447816808552, 9.507796158983861149597414380219, 10.92417432846741936171025278395