L(s) = 1 | + (1.81 + 1.81i)2-s + (2.33 − 1.34i)3-s + 4.59i·4-s + (0.384 + 0.103i)5-s + (6.68 + 1.79i)6-s + (−4.70 + 4.70i)8-s + (2.13 − 3.69i)9-s + (0.511 + 0.885i)10-s + (−3.50 − 0.939i)11-s + (6.19 + 10.7i)12-s + (3.44 − 1.06i)13-s + (1.03 − 0.277i)15-s − 7.90·16-s − 4.09·17-s + (10.5 − 2.83i)18-s + (−0.208 − 0.777i)19-s + ⋯ |
L(s) = 1 | + (1.28 + 1.28i)2-s + (1.34 − 0.778i)3-s + 2.29i·4-s + (0.172 + 0.0461i)5-s + (2.72 + 0.731i)6-s + (−1.66 + 1.66i)8-s + (0.711 − 1.23i)9-s + (0.161 + 0.280i)10-s + (−1.05 − 0.283i)11-s + (1.78 + 3.09i)12-s + (0.955 − 0.294i)13-s + (0.267 − 0.0717i)15-s − 1.97·16-s − 0.993·17-s + (2.49 − 0.668i)18-s + (−0.0478 − 0.178i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(0.259−0.965i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(0.259−0.965i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
0.259−0.965i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(423,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), 0.259−0.965i)
|
Particular Values
L(1) |
≈ |
3.31613+2.54323i |
L(21) |
≈ |
3.31613+2.54323i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+(−3.44+1.06i)T |
good | 2 | 1+(−1.81−1.81i)T+2iT2 |
| 3 | 1+(−2.33+1.34i)T+(1.5−2.59i)T2 |
| 5 | 1+(−0.384−0.103i)T+(4.33+2.5i)T2 |
| 11 | 1+(3.50+0.939i)T+(9.52+5.5i)T2 |
| 17 | 1+4.09T+17T2 |
| 19 | 1+(0.208+0.777i)T+(−16.4+9.5i)T2 |
| 23 | 1−5.09iT−23T2 |
| 29 | 1+(−1.00+1.74i)T+(−14.5−25.1i)T2 |
| 31 | 1+(1.62+6.06i)T+(−26.8+15.5i)T2 |
| 37 | 1+(−1.26+1.26i)T−37iT2 |
| 41 | 1+(−0.578−2.15i)T+(−35.5+20.5i)T2 |
| 43 | 1+(2.65−1.53i)T+(21.5−37.2i)T2 |
| 47 | 1+(−2.19+8.19i)T+(−40.7−23.5i)T2 |
| 53 | 1+(4.54−7.87i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−5.25−5.25i)T+59iT2 |
| 61 | 1+(2.40+1.38i)T+(30.5+52.8i)T2 |
| 67 | 1+(−1.30+4.85i)T+(−58.0−33.5i)T2 |
| 71 | 1+(0.582−2.17i)T+(−61.4−35.5i)T2 |
| 73 | 1+(−4.78+1.28i)T+(63.2−36.5i)T2 |
| 79 | 1+(−1.80−3.13i)T+(−39.5+68.4i)T2 |
| 83 | 1+(−5.36+5.36i)T−83iT2 |
| 89 | 1+(−8.44−8.44i)T+89iT2 |
| 97 | 1+(7.71+2.06i)T+(84.0+48.5i)T2 |
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
−10.96340358541515272339660597473, −9.453429769570230234842552467402, −8.418829332032834294214811568786, −7.968990384633047151251971674431, −7.24110413309657270412541568121, −6.29438476625127345344549865687, −5.50226771329995348111643664469, −4.18412152919424446214248669246, −3.26499389610623280905516094175, −2.27665286795862051792065361959,
1.88953583042435451493249601886, 2.74933686647479726231685512236, 3.64537918731789456518117037835, 4.43419495783935468105548141454, 5.27320176352994121875413294234, 6.52796107807166343396623296517, 8.091383929739466176594723679249, 8.973383141845427840128158937259, 9.780910355502656405736291650069, 10.57620840370255955188848263141