L(s) = 1 | + (1.22 + 2.11i)2-s + (−0.333 + 0.578i)3-s + (−1.99 + 3.45i)4-s + (0.455 + 0.788i)5-s − 1.63·6-s − 4.87·8-s + (1.27 + 2.21i)9-s + (−1.11 + 1.92i)10-s + (−1.83 + 3.18i)11-s + (−1.33 − 2.30i)12-s − 13-s − 0.607·15-s + (−1.97 − 3.42i)16-s + (3.59 − 6.22i)17-s + (−3.12 + 5.41i)18-s + (−0.989 − 1.71i)19-s + ⋯ |
L(s) = 1 | + (0.865 + 1.49i)2-s + (−0.192 + 0.333i)3-s + (−0.997 + 1.72i)4-s + (0.203 + 0.352i)5-s − 0.667·6-s − 1.72·8-s + (0.425 + 0.737i)9-s + (−0.352 + 0.610i)10-s + (−0.554 + 0.960i)11-s + (−0.384 − 0.666i)12-s − 0.277·13-s − 0.156·15-s + (−0.493 − 0.855i)16-s + (0.871 − 1.50i)17-s + (−0.736 + 1.27i)18-s + (−0.226 − 0.392i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.947+0.318i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.947+0.318i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.947+0.318i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(508,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), −0.947+0.318i)
|
Particular Values
L(1) |
≈ |
0.323993−1.98105i |
L(21) |
≈ |
0.323993−1.98105i |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 7 | 1 |
| 13 | 1+T |
good | 2 | 1+(−1.22−2.11i)T+(−1+1.73i)T2 |
| 3 | 1+(0.333−0.578i)T+(−1.5−2.59i)T2 |
| 5 | 1+(−0.455−0.788i)T+(−2.5+4.33i)T2 |
| 11 | 1+(1.83−3.18i)T+(−5.5−9.52i)T2 |
| 17 | 1+(−3.59+6.22i)T+(−8.5−14.7i)T2 |
| 19 | 1+(0.989+1.71i)T+(−9.5+16.4i)T2 |
| 23 | 1+(−0.298−0.516i)T+(−11.5+19.9i)T2 |
| 29 | 1+3.64T+29T2 |
| 31 | 1+(3.54−6.13i)T+(−15.5−26.8i)T2 |
| 37 | 1+(0.355+0.615i)T+(−18.5+32.0i)T2 |
| 41 | 1+5.27T+41T2 |
| 43 | 1−11.0T+43T2 |
| 47 | 1+(−6.05−10.4i)T+(−23.5+40.7i)T2 |
| 53 | 1+(−5.72+9.91i)T+(−26.5−45.8i)T2 |
| 59 | 1+(−4.79+8.30i)T+(−29.5−51.0i)T2 |
| 61 | 1+(−3.49−6.04i)T+(−30.5+52.8i)T2 |
| 67 | 1+(0.614−1.06i)T+(−33.5−58.0i)T2 |
| 71 | 1−11.3T+71T2 |
| 73 | 1+(−3.26+5.65i)T+(−36.5−63.2i)T2 |
| 79 | 1+(−5.76−9.97i)T+(−39.5+68.4i)T2 |
| 83 | 1+7.16T+83T2 |
| 89 | 1+(−6.42−11.1i)T+(−44.5+77.0i)T2 |
| 97 | 1+9.09T+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
−10.99771184634421125170585705967, −10.10241487302533065992884985546, −9.225675585604194263249880332712, −7.940106364815942595924173342021, −7.31466933166626526675765890112, −6.71579981810107386474730949479, −5.31012588225048534054257683097, −5.06868315370243369766181675224, −3.99414902810521792244796202135, −2.55212803839924667043143412570,
0.889740653057662034786035416057, 2.04800182541080329116087102410, 3.43137868067861434165354849843, 4.08859780134221568310393956222, 5.50625150631879163333383535671, 5.91149810246871844407854638259, 7.42144056693408264561366562974, 8.643557519210928867061624461225, 9.590253632991010591466616603021, 10.40394733412138908552717050890