L(s) = 1 | + (0.605 − 1.04i)2-s + (−0.872 − 1.51i)3-s + (0.267 + 0.462i)4-s + (1.10 − 1.91i)5-s − 2.11·6-s + 3.06·8-s + (−0.0222 + 0.0384i)9-s + (−1.33 − 2.31i)10-s + (0.394 + 0.683i)11-s + (0.465 − 0.807i)12-s + 13-s − 3.85·15-s + (1.32 − 2.29i)16-s + (−0.872 − 1.51i)17-s + (0.0268 + 0.0465i)18-s + (2.16 − 3.74i)19-s + ⋯ |
L(s) = 1 | + (0.428 − 0.741i)2-s + (−0.503 − 0.872i)3-s + (0.133 + 0.231i)4-s + (0.494 − 0.856i)5-s − 0.862·6-s + 1.08·8-s + (−0.00740 + 0.0128i)9-s + (−0.423 − 0.733i)10-s + (0.118 + 0.206i)11-s + (0.134 − 0.232i)12-s + 0.277·13-s − 0.995·15-s + (0.330 − 0.573i)16-s + (−0.211 − 0.366i)17-s + (0.00633 + 0.0109i)18-s + (0.495 − 0.858i)19-s + ⋯ |
Λ(s)=(=(637s/2ΓC(s)L(s)(−0.605+0.795i)Λ(2−s)
Λ(s)=(=(637s/2ΓC(s+1/2)L(s)(−0.605+0.795i)Λ(1−s)
Degree: |
2 |
Conductor: |
637
= 72⋅13
|
Sign: |
−0.605+0.795i
|
Analytic conductor: |
5.08647 |
Root analytic conductor: |
2.25532 |
Motivic weight: |
1 |
Rational: |
no |
Arithmetic: |
yes |
Character: |
χ637(79,⋅)
|
Primitive: |
yes
|
Self-dual: |
no
|
Analytic rank: |
0
|
Selberg data: |
(2, 637, ( :1/2), −0.605+0.795i)
|
Particular Values
L(1) |
≈ |
0.863513−1.74191i |
L(21) |
≈ |
0.863513−1.74191i |
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+(−0.605+1.04i)T+(−1−1.73i)T2 |
| 3 | 1+(0.872+1.51i)T+(−1.5+2.59i)T2 |
| 5 | 1+(−1.10+1.91i)T+(−2.5−4.33i)T2 |
| 11 | 1+(−0.394−0.683i)T+(−5.5+9.52i)T2 |
| 17 | 1+(0.872+1.51i)T+(−8.5+14.7i)T2 |
| 19 | 1+(−2.16+3.74i)T+(−9.5−16.4i)T2 |
| 23 | 1+(−0.556+0.963i)T+(−11.5−19.9i)T2 |
| 29 | 1+8.48T+29T2 |
| 31 | 1+(−2.85−4.93i)T+(−15.5+26.8i)T2 |
| 37 | 1+(1.13−1.97i)T+(−18.5−32.0i)T2 |
| 41 | 1+12.1T+41T2 |
| 43 | 1−8.06T+43T2 |
| 47 | 1+(−4.37+7.57i)T+(−23.5−40.7i)T2 |
| 53 | 1+(3.97+6.88i)T+(−26.5+45.8i)T2 |
| 59 | 1+(−5.47−9.48i)T+(−29.5+51.0i)T2 |
| 61 | 1+(6.53−11.3i)T+(−30.5−52.8i)T2 |
| 67 | 1+(3.27+5.67i)T+(−33.5+58.0i)T2 |
| 71 | 1−5.85T+71T2 |
| 73 | 1+(−4.00−6.93i)T+(−36.5+63.2i)T2 |
| 79 | 1+(−3.45+5.98i)T+(−39.5−68.4i)T2 |
| 83 | 1+3.14T+83T2 |
| 89 | 1+(−1.69+2.93i)T+(−44.5−77.0i)T2 |
| 97 | 1−0.0981T+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
−10.52992147417757560736646658669, −9.450651876850462695559479060784, −8.604753431381280732732916535243, −7.38307217759669385539852075185, −6.81335297347388348175700438509, −5.56761446432212740741097057825, −4.70998185669572303304709404730, −3.47822152447108134984713502529, −2.08332049483295383558027868615, −1.08210350213104928818412929754,
1.91307008976626801927001923979, 3.58972026829573545022849499640, 4.61416621282429080495741762374, 5.67122503739849240914972191275, 6.09471675877314802475407713192, 7.12641492194455636591173565909, 7.997065527556914409427668269666, 9.483387305758148563871513835442, 10.10001724661199182003023669768, 10.90391527646709271476664276981