L(s) = 1 | − 2.18i·2-s + (−0.895 − 1.55i)3-s − 2.79·4-s + (1.89 − 1.09i)5-s + (−3.39 + 1.96i)6-s + 1.73i·8-s + (−0.104 + 0.180i)9-s + (−2.39 − 4.14i)10-s + (1.10 − 0.637i)11-s + (2.49 + 4.33i)12-s + (−3.5 + 0.866i)13-s + (−3.39 − 1.96i)15-s − 1.79·16-s − 3·17-s + (0.395 + 0.228i)18-s + (−5.68 − 3.28i)19-s + ⋯ |
L(s) = 1 | − 1.54i·2-s + (−0.517 − 0.895i)3-s − 1.39·4-s + (0.847 − 0.489i)5-s + (−1.38 + 0.800i)6-s + 0.612i·8-s + (−0.0347 + 0.0602i)9-s + (−0.757 − 1.31i)10-s + (0.332 − 0.192i)11-s + (0.721 + 1.24i)12-s + (−0.970 + 0.240i)13-s + (−0.876 − 0.506i)15-s − 0.447·16-s − 0.727·17-s + (0.0932 + 0.0538i)18-s + (−1.30 − 0.753i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.372−0.927i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.372−0.927i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.372−0.927i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(459,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), −0.372−0.927i)
|
Particular Values
L(1) |
≈ |
0.615027+0.910014i |
L(21) |
≈ |
0.615027+0.910014i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(3.5−0.866i)T |
good | 2 | 1+2.18iT−2T2 |
| 3 | 1+(0.895+1.55i)T+(−1.5+2.59i)T2 |
| 5 | 1+(−1.89+1.09i)T+(2.5−4.33i)T2 |
| 11 | 1+(−1.10+0.637i)T+(5.5−9.52i)T2 |
| 17 | 1+3T+17T2 |
| 19 | 1+(5.68+3.28i)T+(9.5+16.4i)T2 |
| 23 | 1−7.58T+23T2 |
| 29 | 1+(−1.10+1.91i)T+(−14.5−25.1i)T2 |
| 31 | 1+(−7.5−4.33i)T+(15.5+26.8i)T2 |
| 37 | 1+6.92iT−37T2 |
| 41 | 1+(2.20+1.27i)T+(20.5+35.5i)T2 |
| 43 | 1+(−2.18−3.78i)T+(−21.5+37.2i)T2 |
| 47 | 1+(−3.70+2.14i)T+(23.5−40.7i)T2 |
| 53 | 1+(−6.08+10.5i)T+(−26.5−45.8i)T2 |
| 59 | 1−8.85iT−59T2 |
| 61 | 1+(−6.37+11.0i)T+(−30.5−52.8i)T2 |
| 67 | 1+(9.87−5.70i)T+(33.5−58.0i)T2 |
| 71 | 1+(0.791−0.456i)T+(35.5−61.4i)T2 |
| 73 | 1+(3+1.73i)T+(36.5+63.2i)T2 |
| 79 | 1+(−3−5.19i)T+(−39.5+68.4i)T2 |
| 83 | 1+3.55iT−83T2 |
| 89 | 1+2.91iT−89T2 |
| 97 | 1+(−13.1+7.61i)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
−10.15930624084380098654332368821, −9.268551201465490186357175743835, −8.732203680607250584980981532085, −7.07829446431214611821892572457, −6.45740792489184069708560120282, −5.16368294417713953081998970209, −4.22720016079006004795848593384, −2.65797190259664534910305746339, −1.78838542180004913324572604439, −0.63068482637995740321499257054,
2.43295887203806432238362271193, 4.32948954678276299851584239643, 4.94130153665576688602015884051, 5.91958148762236628856279437785, 6.56139384277718308642843366832, 7.41797706601794476924588567162, 8.514282798949624798695065891553, 9.390542324136951992824605773174, 10.19885787751744967158820105754, 10.79962877993932160195884515155